What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

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1

Q.

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

Q.

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

Q.

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

Q.

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

Q.

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

By divisible algorithm, when

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
is divided by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
the reminder is a linear polynomial

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
be subtracted from p(x) so that the result is divisible by q(x).

Let

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

We have,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Clearly,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of q(x), therefore, f(x) will be divisible by q(x) if
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
and
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

and

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Adding (i) and (ii), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Putting this value in equation (i), we get,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?

Hence,

What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
will be divisible by
What must be subtracted from X 3 6x 2 15x 80 so that the result is exactly divisible by X 2 X 12 degrees?
if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2