Question

How do you factor the expression `(2x-3)^3 (x+1) +(x-3) (2x-3)^2`?

Answer 1

Combine, simplify and use the difference of squares identity to find:

`(2x-3)^3(x+1)+(x-3)(2x-3)^2`

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

Explanation:

The difference of squares identity can be written:

`a^2-b^2=(a-b)(a+b)`

We use this with `a=x` and `b=sqrt(3)` below...

`(2x-3)^3(x+1)+(x-3)(2x-3)^2`

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

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

`=(2x-3)^2(2x^2-6)`

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

`=2(2x-3)^2(x^2-(sqrt(3))^2)`

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