Question

What must be added to the polynomial $p\left(x\right)=4x^3+8x^2+8x+7$, so that the resulting polynomial is exactly divisible by $2x^2-x+1?$

Answer 1

We divide the polynomial $4x^3+8x^2+8x+7$ by $2x^2-x+1$ as shown in the above image:

From the above division, we conclude that the quotient is $2x+5$ and the remainder is $11x+2$.

Therefore, we should add $-(11x+2)$ to make the given polynomial $4x^3+8x^2+8x+7$ divisible by $2x^2-x+1$.

Hence, $-11x-2$ must be added to the polynomial $4x^3+8x^2+8x+7$ so that the resulting polynomial is exactly divisible by $2x^2-x+1$.