Question

Which of the following polynomials has $(x+2)$ as a factor?

• A. $4x^3-13x+6$
• B. $x^3+x^2+x+4$
• C. $4x^3+13x-25$
• D. $-2x^3+x^2-x-19$

Answer 1

The correct option is A $4x^3-13x+6$

According to the factor theorem, if $(x-a)$ is a factor of $f\left(x\right)$, then $f\left(a\right)=0$.

So, $(x+2)$ will be a factor of a polynomial that satisfies $f(-2)=0$.

Substituting $x=-2$ in each of the given polynomials, we get:

A) $f\left(x\right)=4x^3-13x+6$

$f(-2)=4(-2{)}^3-13(-2)+6=-32+26+6=0$

$\therefore f(-2)=0$

B) $f\left(x\right)=x^3+x^2+x+4$

$f(-2)=(-2{)}^3+(-2{)}^2-2+4=-8+4-2+4=-2$

$\therefore f(-2)=-2$

C) $f\left(x\right)=4x^3+13x-25$

$f(-2)=4(-2{)}^3+13(-2)-25=-32-26-25=-83$

$\therefore f(-2)=-83$

D) $f\left(x\right)={-2x}^3+x^2-x-19$

$f(-2)=-2(-2{)}^3+(-2{)}^2-(-2)-19=16+4+2-19=3$

$\therefore f(-2)=3$

We see that, $f(-2)=0$ is true for only the first polynomial.

Hence, $(x+2)$ is a factor of polynomial given in option A.