# (x + 1) is a factor of the polynomial

Question:

(x + 1) is a factor of the polynomial

(a) $x^{3}+x^{2}-x+1$

(b) $x^{3}+2 x^{2}-x-2$

(c) $x^{3}+2 x^{2}-x+2$

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

Solution:

(b) $x^{3}-2 x^{2}-x-2$

Let:

$f(x)=x^{3}-2 x^{2}-x-2$

By the factor theorem, (x + 1) will be a factor of f(x) if f (-">-1) = 0.
We have:

$f(-1)=(-1)^{3}+2 \times(-1)^{2}-(-1)-2$

$=-1+2+1-2$

$=0$

Hence, $(x+1)$ is a factor of $f(x)=x^{3}+2 x^{2}-x-2$.