# Determine which of the following polynomials has ( (x+1) ) a factor:(i) ( x^{3}+x^{2}+x+1 )(ii) ( x^{4}+x^{3}+x^{2}+x+1 )

Given :

The given term is $(x + 1)$.

The given polynomials are

(i) $x^{3}+x^{2}+x+1$
(ii) $x^{4}+x^{3}+x^{2}+x+1$

To find :

We have to check whether the given polynomials have $(x + 1)$ as a factor.

Solution :

According to factor theorem,

If $(x+1)$ is a factor of given polynomial $P(x)$, then at $x= -1$, $p(x)=0$.

(i) $x^{3}+x^{2}+x+1$

Let $p(x)= x^{3}+x^{2}+x+1$

Substituting $x= -1$

$p(−1)=(−1)^3+(−1)^2+(−1)+1 =−1+1−1+1=0$

Hence, by factor theorem, $x+1$ is a factor of $x^{3}+x^{2}+x+1$.

(ii) $x^{4}+x^{3}+x^{2}+x+1$

Let $p(x)=x^{4}+x^{3}+x^{2}+x+1$

Substituting $x= -1$

$p(−1)=(−1)^4+(-1)^3+(−1)^2+(−1)+1 =1−1+1−1+1=1$

Hence, by factor theorem, $x+1$ is not a factor of $x^{4}+x^{3}+x^{2}+x+1$.

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