Question

How do you factor `27x^3 - 27y^3`?

Answer 1

`(3x-3y)(9x^2 +9xy+9y^2)`

Explanation:

This is the difference of cubes and has a set pattern.

`(3x-3y)(9x^2 +9xy+9y^2)`

make 2 brackets:
The first is simply the cube root of each term

The second bracket is formed from the first in 5 steps:

`1.rarr` square the first term.
`2.rarr` change the sign
`3.rarr` multiply the two terms
`4.rarr` PLUS
`5.rarr`square the second term

Answer 2

`27(x-y)(x^2+xy+y^2)`

Explanation:

The first step here is to take out a `color(blue)"common factor"` of 27.

`rArr27(x^3-y^3)........ (A)`

Now `x^3-y^3` is a `color(blue)"difference of cubes"` and in general factorises as.

`color(red)(bar(ul(|color(white)(a/a)color(black)(a^3-b^3=(a-b)(a^2+ab+b^2))color(white)(a/a)|)))`

here a = x and b = y

`rArrx^3-y^3=(x-y)(x^2+xy+y^2)`

Substituting this result into (A) gives the factorised form.

`rArr27x^3-27y^3=27(x-y)(x^2+xy+y^2)`