# How do you factor completely p(x)=(x^3)-(6x^2)-4x-10?

Nov 16, 2015

$p \left(x\right) = \left(x - {x}_{1}\right) \left(x - {x}_{2}\right) \left(x - {x}_{3}\right)$

where ${x}_{1}$, ${x}_{2}$ and ${x}_{3}$ are as below:

#### Explanation:

$p \left(x\right) = {x}^{3} - 6 {x}^{2} - 4 x - 10$

Attempt to solve $p \left(x\right) = 0$

Let $t = x - 2$

Then:

${t}^{3} - 16 t - 34 = {\left(x - 2\right)}^{3} - 16 \left(x - 2\right) - 34$

$= {x}^{3} - 6 {x}^{2} + 12 x - 8 - 16 x + 32 - 34$

$= {x}^{3} - 6 {x}^{2} - 4 x - 10 = p \left(x\right) = 0$

Let $t = u + v$

$0 = {\left(u + v\right)}^{3} - 16 \left(u + v\right) - 34$

$= {u}^{3} + {v}^{3} + \left(3 u v - 16\right) \left(u + v\right) - 34$

Add the constraint $v = \frac{16}{3 u}$ to make $\left(3 u v - 16\right) = 0$

Then:

$0 = {u}^{3} + {\left(\frac{16}{3 u}\right)}^{3} - 34 = {u}^{3} + \frac{4096}{27 {u}^{3}} - 34$

Multiply both ends by $27 {u}^{3}$ to get:

$0 = 27 {\left({u}^{3}\right)}^{2} - 918 \left({u}^{3}\right) + 4096$

Use the quadratic formula to get:

${u}^{3} = \frac{918 \pm \sqrt{{918}^{2} - \left(4 \cdot 27 \cdot 4096\right)}}{2 \cdot 27}$

$= \frac{918 \pm \sqrt{400396}}{54}$

$= \frac{918 \pm 6 \sqrt{11121}}{54}$

$= \frac{3 \left(153 \pm \sqrt{11121}\right)}{27}$

The derivation is symmetric in $u$ and $v$, hence:

$t = \sqrt[3]{\frac{3 \left(153 + \sqrt{11121}\right)}{27}} + \sqrt[3]{\frac{3 \left(153 - \sqrt{11121}\right)}{27}}$

$= \frac{1}{3} \left(\sqrt[3]{3 \left(153 + \sqrt{11121}\right)} + \sqrt[3]{3 \left(153 - \sqrt{11121}\right)}\right)$

So the Real root of $p \left(x\right) = 0$ is:

${x}_{1} = 2 + t$

$= 2 + \frac{1}{3} \left(\sqrt[3]{3 \left(153 + \sqrt{11121}\right)} + \sqrt[3]{3 \left(153 - \sqrt{11121}\right)}\right)$

The Complex roots are:

${x}_{2} = 2 + \frac{1}{3} \left(\omega \sqrt[3]{3 \left(153 + \sqrt{11121}\right)} + {\omega}^{2} \sqrt[3]{3 \left(153 - \sqrt{11121}\right)}\right)$

${x}_{3} = 2 + \frac{1}{3} \left({\omega}^{2} \sqrt[3]{3 \left(153 + \sqrt{11121}\right)} + \omega \sqrt[3]{3 \left(153 - \sqrt{11121}\right)}\right)$

where $\omega = - \frac{1}{2} + \frac{\sqrt{3}}{2} i = \cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)$ is the primitive Complex cube root of unity.