How do you simplify the expression #(x^4-y^4) /( (x^4+2x^2y^2+y^4)(x^2-2xy+y^2))#?

2 Answers
May 17, 2016

Answer:

#(x+y)/((x^2+y^2)(x-y))#

Explanation:

There are two polynomial equalities that will help us.
#(x-a)(x-a)=x^2-a^2# and #(x + a)^2=x^2+2ax+a^2#
Analyzing the numerator #x^4-y^4 = (x^2+y^2)(x^2-y^2)#
At the denominator we have
#x^4+2x^2y^2+y^4=(x^2+y^2)^2# and #x^2-2xy+y^2=(x-y)^2#
Putting all together
# ((x^2+y^2)(x^2-y^2))/((x^2+y^2)^2(x-y)^2)=(x+y)/((x^2+y^2)(x-y))#

May 17, 2016

Answer:

#(x^4-y^4)/((x^4+2x^2y^2+y^4)(x^2-2xy+y^2))=(x+y)/((x^2+y^2)(x-y))#

Explanation:

#(x^4-y^4)/((x^4+2x^2y^2+y^4)(x^2-2xy+y^2))#

#=((x^2-y^2)color(red)(cancel(color(black)((x^2+y^2)))))/(color(red)(cancel(color(black)((x^2+y^2))))(x^2+y^2)(x-y)(x-y))#

#=(color(red)(cancel(color(black)((x-y))))(x+y))/((x^2+y^2)color(red)(cancel(color(black)((x-y))))(x-y))#

#=(x+y)/((x^2+y^2)(x-y))#

Note that we do not have to specify any exclusions as the values we have cancelled out from the numerator and denominator exist in the denominator of the simplified expression.