Roots of monic polynomials

Question

I'm trying to show that if $\alpha$ is a rational root of $a_nx^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then $a_n \alpha$ is a root of monic $x^n+a_{n-1}x^{n-1}+a_{n}a_{n-2}x^{n-2}+...+a_{n}^{n-2}a_{1}x+a_{n}^{n-1}a_{0}$. I feel like I'm missing something obvious; any help is appreciated.

Answer 1

Hint: If you plug in $a_n \alpha = x$ into the second polynomial, what is the result? Remember that you are given that $a_n \alpha^n + a_{n-1}\alpha^{n-1} + \dots a_1 \alpha + a_0 = 0$