Problem: State the division algorithm for polynomials. Using this result, show that, if the polynomial $f(x)$ has a root $a$, then the linear polynomial $x-a$ divides $f(x)$.
I’m incredibly stuck on this problem from a sample exam. I don’t know what the question wants me to do or even how to answer it. The only thing I know is what the division algorithm is.
Well, from the division algorithm, there exist unique polynomials $q(x)$ and $r(x)$ such that $f(x) = q(x)(x-a) + r(x)$, and $\deg r(x) < \deg (x-1)$.
From the last inequality, note that $\deg(x-1) = 1$ and so $\deg(r(x))=0$, so $r(x) \equiv r$ is a constant. Hence, $$ f(x) = q(x)(x-a) + r. $$
What happens when $a$ is the root of $f$? Can you evaluate this last equation and finish the problem?