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)`.