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

1 Answer
May 4, 2016

Answer:

#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)#