Question

How do you factor `x^3+x^2-125p^3-25p^2`?

Answer 1

`x^3+x^2-125p^3-25p^2 = (x-5p)(x^2+5px+25p^2+x+5p)`

Explanation:

The difference of squares identity can be written:

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

The difference of cubes identity can be written:

`a^3-b^3 = (a-b)(a^2+ab+b^2)`

So we find:

`x^3+x^2-125p^3-25p^2 = (x^3-(5p)^3)+(x^2-(5p)^2)`

`color(white)(x^3+x^2-125p^3-25p^2) = (x-5p)(x^2+5px+25p^2)+(x-5p)(x+5p)`

`color(white)(x^3+x^2-125p^3-25p^2) = (x-5p)(x^2+5px+25p^2+x+5p)`

This has no simpler factors.