Question

How do you factor `x^3 + x^2 -9x -9` by grouping?

Answer 1

(x+1)(x-3)(x+3)

Explanation:

Group the terms into 'pairs' as follows.

`[x^3+x^2]+[-9x-9]`

now factorise each pair.

`color(red)(x^2)(x+1)-color(red)(9)(x+1)`

We now have a common factor of (x+1) which can be 'taken out'

`(x+1)(color(red)(x^2-9))........ (A)`

`color(red)(x^2-9)" is a " color(blue)"difference of squares"` and is factorised in general as follows.

`color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|)))`

`x^2=(x)^2" and " 9=(3)^2rArra=x" and " b=3`

`rArrx^2-9=(x-3)(x+3)`

Substitute these factors into (A)

`rArr(x+1)(x^2-9)=(x+1)(x-3)(x+3)`

Thus `x^3+x^2-9x-9=(x+1)(x-3)(x+3)`