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 $x^{4}+x^{3}+x^{2}+x+1$.