Question

9) If x=-4 is a zero of a polynomial p(x), then the zero of p(x² - 6x + 5) is​

Answer 1

If $x=-4$ is a zero of $p(x)$, then we can write

$p(x) = (x + 4) q(x)$

for another polynomial $q(x)$.

Note that $x^2 - 6x + 5 = (x - 5)(x - 1)$. Then

$p(x^2 - 6x + 5) = p\bigg((x-5)(x-1)\bigg) \\\\ ~~~~~~~~~~~~ = ((x - 5)(x-1) + 4) q\bigg((x-5)(x-1)\bigg) \\\\ ~~~~~~~~~~~~ = (x^2 - 6x + 9) q(x^2 - 6x + 5) \\\\ ~~~~~~~~~~~~ = (x - 3)^2 q(x^2 - 6x + 5)$

so that $\boxed{x = 3}$ is a root of $p(x^2 - 6x + 5)$.