# How do you factor  1 - 2.25x^8?

Jan 3, 2016

$1 - 2.25 {x}^{8}$

$= \frac{1}{4} \left(\sqrt[4]{2} - \sqrt[4]{3} x\right) \left(\sqrt[4]{2} + \sqrt[4]{3} x\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} - \sqrt[4]{24} x + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} + \sqrt[4]{24} x + \sqrt{3} {x}^{2}\right)$

#### Explanation:

Some identities that we will use:

Difference of squares identity:

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

Sum of fourth powers identity:

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

So:

$1 - 2.25 {x}^{8}$

$= \frac{1}{4} \left(4 - 9 {x}^{8}\right)$

$= \frac{1}{4} \left({2}^{2} - {\left(3 {x}^{4}\right)}^{2}\right)$

$= \frac{1}{4} \left(2 - 3 {x}^{4}\right) \left(2 + 3 {x}^{4}\right)$

$= \frac{1}{4} \left({\left(\sqrt{2}\right)}^{2} - {\left(\sqrt{3} {x}^{2}\right)}^{2}\right) \left(2 + 3 {x}^{4}\right)$

$= \frac{1}{4} \left(\sqrt{2} - \sqrt{3} {x}^{2}\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(2 + 3 {x}^{4}\right)$

$= \frac{1}{4} \left({\left(\sqrt[4]{2}\right)}^{2} - {\left(\sqrt[4]{3} x\right)}^{2}\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(2 + 3 {x}^{4}\right)$

$= \frac{1}{4} \left(\sqrt[4]{2} - \sqrt[4]{3} x\right) \left(\sqrt[4]{2} + \sqrt[4]{3} x\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(2 + 3 {x}^{4}\right)$

$= \frac{1}{4} \left(\sqrt[4]{2} - \sqrt[4]{3} x\right) \left(\sqrt[4]{2} + \sqrt[4]{3} x\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left({\left(\sqrt[4]{2}\right)}^{4} + {\left(\sqrt[4]{3} x\right)}^{4}\right)$

$= \frac{1}{4} \left(\sqrt[4]{2} - \sqrt[4]{3} x\right) \left(\sqrt[4]{2} + \sqrt[4]{3} x\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} - \sqrt{2} \sqrt[4]{2} \sqrt[4]{3} x + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} + \sqrt{2} \sqrt[4]{2} \sqrt[4]{3} x + \sqrt{3} {x}^{2}\right)$

$= \frac{1}{4} \left(\sqrt[4]{2} - \sqrt[4]{3} x\right) \left(\sqrt[4]{2} + \sqrt[4]{3} x\right) \left(\sqrt{2} + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} - \sqrt[4]{24} x + \sqrt{3} {x}^{2}\right) \left(\sqrt{2} + \sqrt[4]{24} x + \sqrt{3} {x}^{2}\right)$