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

Feb 19, 2016

$x = 1 , i , - i$

#### Explanation:

Step 1: Set the equation to equal to zero

${x}^{3} - {x}^{2} + x - 1 = 0$

Step 2: Group the terms like this

$\textcolor{red}{\left({x}^{3} - {x}^{2}\right)} + \textcolor{b l u e}{\left(x - 1\right)} = 0$

Step 3: Factor out the common like term for each group

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

Step 4 Set each factor equal to zero

${x}^{2} + 1 \text{ } = 0$
$\text{ " -1 " " " } - 1$
${x}^{2} \text{ " " = " } - 1$
$\sqrt{{x}^{2}} \text{ } = \sqrt{- 1}$

$x = \pm i$

or

$x - 1 = 0$
$x = 1$

Answer: $x = 1 , i , - i$