How do you factor $x^{3} + x^{2} + x + 1$ $x^3+x^2+x+1$ ?

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How do you factor $x^{3} + x^{2} + x + 1$ $x^3+x^2+x+1$ ?

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Answer 1 · 2018-04-24T10:27:42

$(x + 1) (x + i) (x - i)$ $(x+1)(x+i)(x-i)$

Explanation:

$factor by grouping$ $color(blue)"factor by grouping"$

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

$take out the common factor (x + 1)$ $"take out the "color(blue)"common factor "(x+1)$

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

$we can factor x^{2} + 1 by solving x^{2} + 1 = 0$ $"we can factor "x^2+1" by solving "x^2+1=0$

$x^{2} + 1 = 0 \Rightarrow x^{2} = - 1 \Rightarrow x = \pm i$ $x^2+1=0rArrx^2=-1rArrx=+-i$

$\Rightarrow x^{2} + 1 = (x + i) (x - i)$ $rArrx^2+1=(x+i)(x-i)$

$\Rightarrow x^{3} + x^{2} + x + 1 = (x + 1 (x + i) (x - i)$ $rArrx^3+x^2+x+1=(x+1(x+i)(x-i)$

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