Question

How do you factor the trinomial `a^3-2a^2-63a`?

Answer 1

`a(a+7)(a-9)`

Explanation:

Given
`color(white)("XXX")a^3-2a^2-63a`

First extract the obvious common factor of `a`
`color(white)("XXX")a(a^2-2a-63)`

We could factor `(a^2-2a-63)` using the quadratic formula
but it would be more common to try to factor a bit more directly:

We are looking for two numbers such that
`color(white)("XXX")`one is positive and one is negative
`color(white)("XXX")`their product is `-63`
`color(white)("XXX")`their sum is `-2`

Noting that `63=7xx3xx3`
we can quickly find the numbers `-9` and `7`

giving the complete factorization:
`color(white)("XXX")a(a-9)(a+7)`