Question

How do you factor completely `−12x^3y^5 − 9x^2y^2 + 12xy^3`?

Answer 1

`-12x^3y^5-9x^2y^2+12xy^3 = -3xy^2(4x^2y^3+3x-4y)`

But note that:

`color(red)(9x^4y^4)-12x^3y^5-9x^2y^2+12xy^3 = 3xy^2(xy-1)(xy+1)(3x-4y)`

Explanation:

Given:

`-12x^3y^5-9x^2y^2+12xy^3`

First note that all of the terms are divisible by `3`, `x` and `y^2`, and hence by `3xy^2`. So we can separate that out as a factor. I will use `-3xy^2` because I prefer positive leading coefficients:

`-12x^3y^5-9x^2y^2+12xy^3 = -3xy^2(4x^2y^3+3x-4y)`

It is not possible to factor this further.

I suspect that a leading term `9x^4y^4` is missing from the question, since we find:

`9x^4y^4-12x^3y^5-9x^2y^2+12xy^3 = 3xy^2(3x^3y^2-4x^2y^3-3x+4y)`

`color(white)(9x^4y^4-12x^3y^5-9x^2y^2+12xy^3) = 3xy^2((3x^3y^2-4x^2y^3)-(3x-4y))`

`color(white)(9x^4y^4-12x^3y^5-9x^2y^2+12xy^3) = 3xy^2(x^2y^2(3x-4y)-1(3x-4y))`

`color(white)(9x^4y^4-12x^3y^5-9x^2y^2+12xy^3) = 3xy^2(x^2y^2-1)(3x-4y)`

`color(white)(9x^4y^4-12x^3y^5-9x^2y^2+12xy^3) = 3xy^2((xy)^2-1^2)(3x-4y)`

`color(white)(9x^4y^4-12x^3y^5-9x^2y^2+12xy^3) = 3xy^2(xy-1)(xy+1)(3x-4y)`