Question

$x+1$ is a factor of the polynomial
$A.$ $x^3+x^2-x+1$
$B.$ $x^3+x^2+x+1$
$C.$ $x^4+x^3+x^2+1$
$D.$ ​​​​​​​ $x^4+3x^3+3x^2+x+1$​​​​​​​

Answer 1

The answer is B. $x^3+x^2+x+1$

Factor theorem: If $(x-a)$ is factor of $P\left(x\right)$ then $P\left(a\right)=0$.

Lets check each option whether $(x+1)$ is te factor of the given polynomial or not.

Option $A:$

Let assume $(x+1)$ is a factor of $x^3+x^2-x+1$

So, $x=-1$ is zero of $x^3+x^2-x+1$

$\Rightarrow (-1{)}^3+(-1{)}^2-(-1)+1$

$=-1+1+1=-1\ne 0$

Hence, Option $A$ is incorrect.

Option $B:$
Let assume $(x+1)$ is a factor of $x^3+x^2+x+1$
So, $x=-1$ is zero of $x^3+x^2+x+1$
$\Rightarrow (-1{)}^3+(-1{)}^2+(-1)+1$
$=-1+1-1+1=0$
Hence, Option $B$ correct.

Option $C:$

Let assume $(x+1)$ is a factor of $x^4+x^3+x^2+1$

So, $x=-1$ is zero of $x^4+x^3+x^2+1$

$\Rightarrow (-1{)}^4+(-1{)}^3+(-1{)}^2+1$

$=1-1+1+1=2\ne 0$

Hence, Option $C$ is incorrect.

Option $D:$

Let assume $(x+1)$ is a factor of $x^4+3x^3+3x^2+x+1$​​​​​​​

So, $x=-1$ is zero of $x^4+3x^3+3x^2+x+1$​​​​​​​

$\Rightarrow (-1{)}^4+3(-1{)}^3+3(-1{)}^2+(-1)+1$

$=1-3+3-1+1=1\ne 0$

Hence, Option $D$ is incorrect.