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

Question

Kennedy Adams · Accepted Answer

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

Ethan Adkins · 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))#

#(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.

Savannah Anderson · Answer

undefined To simplify the expression (x^4-y^4) /( (x^4+2x^2y^2+y^4)(x^2-2xy+y^2)), we can factor the numerator and denominator. The numerator can be factored as (x^2+y^2)(x^2-y^2), which further simplifies to (x^2+y^2)(x+y)(x-y). The denominator can be factored as (x^2+y^2)(x-y)^2.

By canceling out the common factors of (x^2+y^2) and (x-y), the expression simplifies to (x+y)/(x-y)^2.