Determine if the polynomial has $(x + 1)$ as a factor: $x^{3} + x^{2} + x + 1$

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Solution

Factor of polynomial.
If $(x + a)$ is a factor of a given polynomial $P (x)$ . Then, $P (- a) = 0$ (Factor theorem)
Here, $P (x) = x^{3} + x^{2} + x + 1$
Need to check if $(x + 1)$ is a factor of $x^{3} + x^{2} + x + 1$ or not.
Substitute $x = - 1$ in the given polynomial:
$P (- 1) = {(- 1)}^{3} + {(- 1)}^{2} + (- 1) + 1 \Rightarrow P (- 1) = - 1 + 1 - 1 + 1 \Rightarrow P (- 1) = 0$
Hence, $(x + 1)$ is a factor of the given polynomial $P (x) = x^{3} + x^{2} + x + 1$ .

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