# How do you use the factor theorem to determine whether x+1 is a factor of  x^3 + x^2 + x + 1?

##### 1 Answer
Jan 3, 2016

The factor theorem states that a polynomial $f \left(x\right)$ has a factor $\left(x + k\right)$ if and only if $f \left(- k\right) = 0$.
Here ${x}^{3} + {x}^{2} + x + 1$ is a polynomial.
Let $f \left(x\right) = {x}^{3} + {x}^{2} + x + 1$

Now we want to know that is $x + 1$ a factor of $f \left(x\right)$ or not.

For this purpose we have to put$x = - 1$ in $f \left(x\right)$, if the result comes to be $0$ then $x + 1$ is a factor of $f \left(x\right)$ and if the result comes not to be $0$ then $x + 1$ is not a factor of $f \left(x\right)$.

Put $x = - 1$ in $f \left(x\right)$
$\implies f \left(- 1\right) = {\left(- 1\right)}^{3} + {\left(- 1\right)}^{2} + \left(- 1\right) + 1 = - 1 + 1 - 1 + 1 = 0$

Since the result is $0$, therefore $x + 1$ is a factor of the given polynomial.