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