Determine $(x + 1)$ is a factor of $x^{4} + x^{3} + x^{2} + x + 1$ or not.

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Solution

Applying remainder theorem, we get
$x + 1 = 0$
$x = - 1$

Now,
$x^{4} + x^{3} + x^{2} + x + 1$

Replacing $x$ with $- 1$ , we get
$= (- 1)^{4} + (- 1)^{3} + (- 1)^{2} + (- 1) + 1$
$= 1 - 1 + 1 - 1 + 1$
$= 1$

Hence, $(x + 1)$ is not a factor of polynomial $(x^{4} + x^{3} + x^{2} + x + 1)$

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