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

Aug 10, 2016

${\left(a + b\right)}^{6} - {\left(a - b\right)}^{6} = 4 a b \left(3 {a}^{2} + {b}^{2}\right) \left({a}^{2} + 3 {b}^{2}\right)$

#### Explanation:

The difference of squares identity can be written:

${x}^{2} - {y}^{2} = \left(x - y\right) \left(x + y\right)$

The difference of cubes identity can be written:

${x}^{3} - {y}^{3} = \left(x - y\right) \left({x}^{2} + x y + {y}^{2}\right)$

The sum of cubes identity can be written:

${x}^{3} + {y}^{3} = \left(x + y\right) \left({x}^{2} - x y + {y}^{2}\right)$

Hence:

${x}^{6} - {y}^{6}$

$= {\left({x}^{3}\right)}^{2} - {\left({y}^{3}\right)}^{2}$

$= \left({x}^{3} - {y}^{3}\right) \left({x}^{3} + {y}^{3}\right)$

$= \left(x - y\right) \left({x}^{2} + x y + {y}^{2}\right) \left(x + y\right) \left({x}^{2} - x y + {y}^{2}\right)$

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

${\left(a + b\right)}^{6} - {\left(a - b\right)}^{6}$

$= {x}^{6} - {y}^{6}$

$= \left(x - y\right) \left({x}^{2} + x y + {y}^{2}\right) \left(x + y\right) \left({x}^{2} - x y + {y}^{2}\right)$

$= \left(\left(a + b\right) - \left(a - b\right)\right) \left({\left(a + b\right)}^{2} + \left(a + b\right) \left(a - b\right) + {\left(a - b\right)}^{2}\right) \left(\left(a + b\right) + \left(a - b\right)\right) \left({\left(a + b\right)}^{2} - \left(a + b\right) \left(a - b\right) + {\left(a - b\right)}^{2}\right)$

$= \left(2 b\right) \left({a}^{2} + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 a b}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{{b}^{2}}}} + {a}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{b}^{2}}}} + {a}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{2 a b}}} + {b}^{2}\right) \left(2 a\right) \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{{a}^{2}}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 a b}}} + {b}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{a}^{2}}}} + {b}^{2} + {a}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{2 a b}}} + {b}^{2}\right)$

$= 4 a b \left(3 {a}^{2} + {b}^{2}\right) \left({a}^{2} + 3 {b}^{2}\right)$