Question

How do you factor `x^3 + 6x^2 - 9x - 54` by grouping?

Answer 1

(x+6)(x-3)(x+3)

Explanation:

Group the terms into 'pairs' as follows.

`[x^3+6x^2]+[-9x-54]`

Now, factorise each 'pair'

`color(red)(x^2)(x+6)color(red)(-9)(x+6)`

Take out the common factor of (x +6)

`(x+6)color(red)((x^2-9))`

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

`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`

`rArrcolor(red)(x^2-9)=(x-3)(x+3)`

Pulling this altogether we get.

`x^3+6x^2-9x-54=(x+6)(x-3)(x+3)`