How do you factor #x^(4n) - y^(4n)#?

1 Answer
May 12, 2016

Answer:

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

Explanation:

Note that both #x^(4n)# and #y^(4n)# are squared terms:

  • #x^(4n)=(x^(2n))^2#
  • #y^(4n)=(y^(2n))^2#

With this knowledge, we will factor this expression as a difference of squares:

  • #color(red)a^2-color(blue)b^2=(color(red)a+color(blue)b)(color(red)a-color(blue)b)#

Thus, we see that

#x^(4n)-y^(4n)#

#=(color(red)(x^(2n)))^2-(color(blue)(y^(2n)))^2#

#=(color(red)(x^(2n))+color(blue)(y^(2n)))(color(red)(x^(2n))-color(blue)(y^(2n)))#

Notice that we can factor #(x^(2n)-y^(2n))# in the same way:

#x^(4n)-y^(4n)=(x^(2n)+y^(2n))(x^(2n)-y^(2n))#

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

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