Question

How do you factor given that `f(-5)=0` and `f(x)=x^3-x^2-21x+45`?

Answer 1

`(x+5)(x-3)^2`

Explanation:

`f(x)=x^3-x^2-21x+45` and `f(-5)=0`

`f(-5)=0` indicates that one of the factors is `(x+5)` and the zero is `-5`.

Use synthetic division with the given zero of `-5`.

`-5color(white)(a)|1color(white)(aa)-1color(white)(a)-21color(white)(a aa^1)45`
`color(white)(aaaa^1)darrcolor(white)(a)-5color(white)(aa^1a)30color(white)a-45`
`color(white)(aaaaa^1)1color(white)(a^1)-6color(white)(aaaa)9color(white)(aaaaa)0`

Use the result of synthetic division as the coefficients of a new polynomial, i.e `(x^3-x^2-21x+4)-:(x+5)=(x^2-6x+9)`.

`1x^2-6x+9`

`(x-3)(x-3)color(white)(aaa)`Factor

The complete factorization is

`(x+5)(x-3)(x-3)=(x+5)(x-3)^2`