Question

How do you factor `30x^5y-85x^3y^2+25xy^3`?

Answer 1

`5xy(y - 2x^2)(y - 3x^2)`

Explanation:

`F(x, y) = 5xyf(x, y) = 5xy(5y^2 - 17 x^2y + 6x^4).`
Factor f(x, y) by considering y as variable and x as constant.
Use the new AC Method to factor trinomials (Socratic Search).
`f(x, y) = 5y^2 - 17x^2y + 6x^4 =` 5(y + p)(y + q)
Converted trinomial: `f'(x, y) = y^2 - 17x^2y + 30x^4 =` (y + p')(y + q')
p' and q' have same sign because ac > 0.
Factor pairs of `(ac = 30x^4)` --> `(-2x^2, -15x^2)`. This sum is `-17x^2 = b`. Therefor, `p' = -2x^2` and `q' = -15x^2`.
Back to original f(x, y) --> `p = (p')/a = -(2x^2)/5` and
`q = (q')/a = -(15x^2)/5 = -3x^2`
Factored form of `f(x) = 5(y - (2x^2)/5)(y - 3x^2) = (5y - 2x^2)(y - 3x^2)`
`F(x) = 5xy.f(x) = 5xy(y - 2x^2)(y - 3x^2)`