$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Question

Given the polynomials

$P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$

such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$.

I would normally apply little Bézout's theorem here and get a system of three equations which I would solve for $a$, $b$ and $c$, but I can't find the roots of $Q(x)$, nor factor it for that matter. Please help.

Answer 1

Start dividing $Q(x)$ into $P(x)$.

$$P(x)-xQ(x)=-3x^3+(a+3)x^2+(b-1)x+c\;,$$ getting rid of the fourth power, so $D(x)=x-3$. Now

$$D(x)Q(x)=(x-3)(x^3+2x^2-3x+1)=x^4-x^3-9x^2+10x-3$$

(assuming no mental algebraic errors on my part), and you can equate coefficients in

$$R(x)=P(x)-D(x)Q(x)$$

to find $a,b$, and $c$.