## A fact about the roots of monic polynomials

Let $p(x) = \sum_{i=0}^n a_ix^i$. Show that if $\alpha$ is a root of $p(x)$, then $a_n\alpha$ is a root of the monic polynomial $q(x) = x^n + \sum_{i=0}^{n-1} a_n^{n-1-i}a_ix^i$.

Note that $q(a_n\alpha) = a_n^n\alpha^n + \sum_{i=0}^{n-1} a_n^{n-1}a_i\alpha^i$ $= a_n^{n-1} p(\alpha) = 0$, as desired.