# How do you find all the zeros of x^3-x^2+2x+1?

May 18, 2016

Use Cardano's method to find Real zero:

${x}_{1} = \frac{1}{3} \left(1 + \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}\right)$

and Complex zeros.

#### Explanation:

Premultiply by ${3}^{3}$ to cut down on arithmetic involving fractions:

$0 = {3}^{3} \left({x}^{3} - {x}^{2} + 2 x + 1\right)$

$= 27 {x}^{3} - 27 {x}^{2} + 54 x + 27$

$= {\left(3 x - 1\right)}^{3} + 15 \left(3 x - 1\right) + 43$

Let $t = 3 x - 1$ and solve:

${t}^{3} + 15 t + 43 = 0$

Using Cardano's method, let $t = u + v$

${u}^{3} + {v}^{3} + 3 \left(u v + 5\right) \left(u + v\right) + 43 = 0$

Let $v = - \frac{5}{u}$ to eliminate the term in $\left(u + v\right)$

${u}^{3} - {5}^{3} / {u}^{3} + 43 = 0$

Multiply through by ${u}^{3}$ to get a quadratic in ${u}^{3}$:

${\left({u}^{3}\right)}^{2} + 43 \left({u}^{3}\right) - 125 = 0$

Use the quadratic formula to find:

${u}^{3} = \frac{- 43 \pm \sqrt{{43}^{2} + \left(4 \cdot 125\right)}}{2}$

$= \frac{- 43 \pm \sqrt{1849 + 500}}{2}$

$= \frac{- 43 \pm \sqrt{2349}}{2}$

$= \frac{- 43 \pm 9 \sqrt{29}}{2}$

The derivation was symmetric in $u$ and $v$, so we can use one of these roots for ${u}^{3}$ and the other for ${v}^{3}$ to find the Real root of our cubic equation in $t$ is:

${t}_{1} = \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}$

and Complex roots:

${t}_{2} = \omega \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + {\omega}^{2} \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}$

${t}_{3} = {\omega}^{2} \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + \omega \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}$

where $\omega = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ is the primitive Complex cube root of $1$.

Then $x = \frac{1}{3} \left(t + 1\right)$, hence zeros of the original cubic in $x$:

${x}_{1} = \frac{1}{3} \left(1 + \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}\right)$

${x}_{2} = \frac{1}{3} \left(1 + \omega \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + {\omega}^{2} \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}\right)$

${x}_{3} = \frac{1}{3} \left(1 + {\omega}^{2} \sqrt[3]{\frac{- 43 + 9 \sqrt{29}}{2}} + \omega \sqrt[3]{\frac{- 43 - 9 \sqrt{29}}{2}}\right)$