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

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

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Answer 1 · 2016-02-19T19:37:49

$x = 1, i, - i$ $x = 1 , i , -i$

Explanation:

Step 1: Set the equation to equal to zero

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

Step 2: Group the terms like this

$(x^{3} - x^{2}) + (x - 1) = 0$ $color (red)((x^3-x^2)) + color(blue)((x-1)) = 0$

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

$x^{2} (x - 1) + 1 (x - 1) = 0$ $x^2color(red)((x-1)) + 1color(red)((x-1)) = 0$
$(x^{2} + 1) (x - 1) = 0$ $(x^2 +1)(x-1) = 0$

Step 4 Set each factor equal to zero

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

$x = \pm i$ $x = +- i$

or

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

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

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

1 Answer

Explanation:

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