$Check whether x^{4} + 3 x^{3} + 3 x^{2} + x + 1 has (x + 1) as a factor.$

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Solution

$Let p (x) = x^{4} + 3 x^{3} + 3 x^{2} + x + 1$
Now the zero of x+1 is -1.
$p (- 1) = (- 1)^{4} + 3 (- 1)^{3} + 3 (- 1)^{2} + (- 1) + 1$
$= 1 - 3 + 3 - 1 + 1$
$= 1 \neq 0$
$By factor theorem, x + 1 is not a factor of x^{4} + 3 x^{3} + 3 x^{2} + x + 1.$

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