What must be added to the polynomial x4 2x3 2x2 x 1 so that the resulting polynomial is exactly divisible by x2 2x 3?

We know that,

f(x) = g(x) x q(x) + r(x)

f(x) - r(x) = g(x) x q(x)

f(x) + {- r(x)} = g(x) x q(x)

Clearly , Right hand side is divisible by g(x).

Therefore, Left hand side is also divisible by g(x).Thus, if we add - r(x) to f(x), then the resulting polynomial is divisible by g(x).

Let us now find the remainder when f(x) is divided by g(x).

Hence, we should add - r(x) = x - 2 to f(x) so that the resulting polynomial is divisible by g(x).

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We know that Dividend = Quotient x Divisor + Remainder.

Dividend - Remainder = Quotient x Divisor.

Clearly, Right hand side of the above result is divisible by the divisor.

Therefore, left hand side is also divisible by the divisor.

Thus, if we subtract remainder from the dividend, then it will be exactly divisible by the divisor.

Dividing x4 + 2x3 − 13x2 − 12x + 21 by x2 − 4x + 3

Therefore, quotient = x2 + 6x + 8 and remainder = (2x - 3).

Thus, if we subtract the remainder 2x - 3 from x4 + 2x3 − 13x2 − 12x + 21 it will be divisible by x2 − 4x + 3.