Determine which of the following polynomials has (x + 1) a factor: <br/> <br/>(i) $x^{3}+x^{2}+x+1$<br/> <br/> (ii) $x^{4}+x^{3}+x^{2}+x+1$<br/> <br/> (iii) $x^{4}+3 x^{3}+3 x^{2}+x+1$ <br/> <br/>(iv) $x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}$

Solution:

(i) If $(x+1)$ is a factor of $p(x)=x^{3}+x^{2}+x+1$, then $p(-1)$ must be zero, otherwise $(x+1)$ is not a factor of $p(x)$.

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

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

$=-1+1-1+1=0$

Hence, $x+1$ is a factor of this polynomial.

(ii) If $(x+1)$ is a factor of $p(x)=x^{4}+x^{3}+x^{2}+x+1$, then $p(-1)$ must be zero, otherwise $(x+1)$ is not a factor of $p(x)$.

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

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

$=1-1+1-1+1=1$

As $p(-1) \neq 0$

Therefore, $x+1$ is not a factor of this polynomial.

(iii) If $(x+1)$ is a factor of polynomial $p(x)=x^{4}+3 x^{3}+3 x^{2}+x+1$, then $p(-1)$ must be 0 , otherwise $(x+1)$ is not a factor of this polynomial.

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

$=1-3+3-1+1=1$

$\operatorname{As} p(-1) \neq 0$

Therefore, $x+1$ is not a factor of this polynomial.

(iv) If $(x+1)$ is a factor of polynomial $p(x)=x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}$, then $p(-1)$ must be 0, otherwise $(x+1)$ is not a factor of this polynomial.

$p(-1)=(-1)^{3}-(-1)^{2}-(2+\sqrt{2})(-1)+\sqrt{2}$

$=-1-1+2+\sqrt{2}+\sqrt{2}$

$=2 \sqrt{2}$

As $p(-1) \neq 0$

Therefore, $(x+1)$ is not a factor of this polynomial.