If (x-1) is a factor of $x^{4} + x^{3} - 2 x^{2} + x + 1$ . Find the other factor using the factor theorem.

$x^{3} + 2 x^{2} + x + 1$

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$x^{3} + x^{2} + x$

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$x^{3} + 2 x^{2} + 1$

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$x^{3} + 2 x^{2} + 1$

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Solution

The correct option is D
$x^{3} + 2 x^{2} + 1$

Since $x - 1$ is a factor of $x^{4} + x^{3} - 2 x^{2} + x + 1$ , we can find the other factor by dividing $x^{4} + x^{3} - 2 x^{2} + x + 1$ by $x - 1$

$\begin{matrix} x^{3} + 2 x^{2} + 1 x - 1 x^{4} + x^{3} - 2 x^{2} + x - 1 x^{4} - x^{3} - -------------- - 2 x^{3} - 2 x^{2} 2 x^{3} - 2 x^{2} - ---------------- - x - 1 x - 1 - ---- - 0 \end{matrix}$

$∴$ $x^{4} + x^{3} - 2 x^{2} + x - 1$ = $(x - 1) (x^{3} + 2 x^{2} + 1)$

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If (x-1) is a factor of x4+x3−2x2+x+1. Find the other factor using the factor theorem.

If (x-1) is a factor of $x^{4} + x^{3} - 2 x^{2} + x + 1$ . Find the other factor using the factor theorem.