Question

How do you factor `x^3 (2x - y) + xy(2x - y)^2`?

Answer 1

`x^3(2x-y)+xy(2x-y)^2=(2x-y)x(x-(1+sqrt(2))y)(x-(1-sqrt(2))y)`

Explanation:

Use the difference of squares identity:

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

with `a=(x-y)` and `b=sqrt(2)y` as follows:

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

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

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

`=(2x-y)x(x^2-2xy-y^2)`

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

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

`=(2x-y)x((x-y)-sqrt(2)y)((x-y)+sqrt(2)y)`

`=(2x-y)x(x-(1+sqrt(2))y)(x-(1-sqrt(2))y)`