How do you factor #(a+b)^6 - (a-b)^6#?

1 Answer
George C.
Aug 10, 2016

#(a+b)^6-(a-b)^6=4ab(3a^2+b^2)(a^2+3b^2)#

Explanation:

The difference of squares identity can be written:

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

The difference of cubes identity can be written:

#x^3-y^3=(x-y)(x^2+xy+y^2)#

The sum of cubes identity can be written:

#x^3+y^3=(x+y)(x^2-xy+y^2)#

Hence:

#x^6-y^6#

#=(x^3)^2-(y^3)^2#

#=(x^3-y^3)(x^3+y^3)#

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

Now let #x=a+b# and #y=a-b# to find:

#(a+b)^6-(a-b)^6#

#= x^6-y^6#

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

#=((a+b)-(a-b))((a+b)^2+(a+b)(a-b)+(a-b)^2)((a+b)+(a-b))((a+b)^2-(a+b)(a-b)+(a-b)^2)#

#=(2b)(a^2+color(red)(cancel(color(black)(2ab)))+color(red)(cancel(color(black)(b^2)))+a^2-color(red)(cancel(color(black)(b^2)))+a^2-color(red)(cancel(color(black)(2ab)))+b^2)(2a)(color(red)(cancel(color(black)(a^2)))+color(red)(cancel(color(black)(2ab)))+b^2-color(red)(cancel(color(black)(a^2)))+b^2+a^2-color(red)(cancel(color(black)(2ab)))+b^2)#

#=4ab(3a^2+b^2)(a^2+3b^2)#

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