How do you find all zeros for $x^{3} - 5 x^{2} + x - 5$ $x^3 - 5x^2 + x - 5$ ?

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How do you find all zeros for $x^{3} - 5 x^{2} + x - 5$ $x^3 - 5x^2 + x - 5$ ?

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Answer 1 · 2016-05-17T15:32:26

$x = 5$ $x =5$
or
$x = \pm i$ $x=+-i$

Explanation:

$x^{3} - 5 x^{2} + x - 5$ $x^3-5x^2+x-5$ can be factored as
$XXX (x^{3} + x) - (5 x^{2} + 5)$ $color(white)("XXX")color(red)(""(x^3+x))-color(blue)(""(5x^2+5))$

$XXX = x (x^{2} + 1) - 5 (x^{2} + 1)$ $color(white)("XXX")=color(red)(x(x^2+1))-color(blue)(5(x^2+1)$

$XXX = (x - 5) \cdot (x^{2} + 1)$ $color(white)("XXX")=(color(red)(x)-color(blue)(5))*(x^2+1)$

For the zeroes:
either
$XXX (x - 5) = 0 XX \to XX x = 5$ $color(white)("XXX")(x-5)=0color(white)("XX")rarrcolor(white)("XX")x=5$
or
$XXX (x^{2} + 1) = 0 XX \to XX x = \pm \sqrt{- 1} = \pm i$ $color(white)("XXX")(x^2+1)=0color(white)("XX")rarrcolor(white)("XX")x=+-sqrt(-1)=+-i$

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How do you find all zeros for $x^{3} - 5 x^{2} + x - 5$ $x^3 - 5x^2 + x - 5$ ?

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How do you find all zeros for x^3 - 5x^2 + x - 5x3−5x2+x−5x^3 - 5x^2 + x - 5?

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How do you find all zeros for $x^{3} - 5 x^{2} + x - 5$ $x^3 - 5x^2 + x - 5$ ?