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

1 Answer
May 13, 2016

Answer:

#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)#