Problem Regarding Polynomial and Division Algorithm

Question

Divide the polynomial $ P(x) = x^4+3x^3-7x^2+11x-1 $ by $ x^2+2 $ and write your result in the form of $ P(x) = (x^2+2)Q(x)+cx+d $.

Thanks!

Answer 1

Let $a$ be a root of $x^2+2=0$. Then $a^2=-2$ and we have:

$$P(a) = ca+d$$

but $$P(a) = 4-6a+14+11a-1 = 5a+17$$

so, since $a$ is pure imaginary we have $$5a+17 = ca+d\implies c=5, d= 17$$