# How do you factor a^8 - a^2b^6?

May 10, 2016

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

#### Explanation:

${a}^{2 + 6} - {a}^{2} {b}^{6}$

$= {a}^{2} {a}^{6} - {a}^{2} {b}^{6}$

$= {a}^{2} \left({a}^{6} - {b}^{6}\right)$, here $\textcolor{red}{{a}^{2}}$ is common between the terms

$= {a}^{2} \left({a}^{3 \times 2} - {b}^{3 \times 2}\right)$

$= {a}^{2} \left[{\left({a}^{3}\right)}^{2} - {\left({b}^{3}\right)}^{2}\right]$

$\textcolor{red}{\text{This is of the form " x^2-y^2 = (x-y)(x+y), "where " x=a^3 " and } y = {b}^{3}}$

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

Now we factorize $\left({a}^{3} - {b}^{3}\right)$ and $\left({a}^{3} + {b}^{3}\right)$:

color(red)("We know that " (a^3-b^3)=(a-b)(a^2+ab+b^2)

color(red)("And "(a^3+b^3)=(a+b)(a^2-ab+b^2)

Then,

a^2(a^3-b^3)(a^3+b^3)=color(blue)(a^2(a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)