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.

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