Question

Which of the following is true regarding the polynomial $f\left(x\right)=x^3-2x^2-x+2$ ?
(Use Factor Theorem)

• A. $(x-1)$ is a factor of $f\left(x\right)$
• B. $(x-2)$ is a factor of $f\left(x\right)$
• C. $(x+2)$ is a factor of $f\left(x\right)$
• D. All of the above

Answer 1

Answer: (A), (B)

$(x-2)$ is a factor of $f\left(x\right)$

Given polynomial is $f\left(x\right)=x^3-2x^2-x+2$.

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

$f\left(1\right)=(1{)}^3-2(1{)}^2-\left(1\right)+2=0$
$\Rightarrow (x-1)$ is a factor of $f\left(x\right)$

$f\left(2\right)=(2{)}^3-2(2{)}^2-\left(2\right)+2=0$
$\Rightarrow (x-2)$ is a factor of $f\left(x\right)$

$f(-2)=(-2{)}^3-2(-2{)}^2-(-2)+2=-12\ne 0$
$\Rightarrow (x+2)$ is not a factor of $f\left(x\right)$