Question

Determine which of the following polynomials has $(x-2)$ a factor :

$3x^2+6x-24$

Answer 1

Determining whether polynomial has $(x-2)$ a factor or not

Let $p\left(x\right)=3x^2+6x-24$ be the polynomial

By factor theorem

If $p\left(x\right)$ is a polynomial of degree $n>1$ and $a$ is any real number, then $x–a$ is a factor of $p\left(x\right)$, if $p\left(a\right)=0$,

Therefore, $(x-2)$ is a factor of $p\left(x\right)$, if $p\left(2\right)=0$

$p\left(x\right)=3x^2+6x-24$

$\Rightarrow p\left(2\right)=3\times {\left(2\right)}^2+6\times 2-24$

$\Rightarrow p\left(2\right)=12+12-24$

$\Rightarrow p\left(2\right)=24-24=0$

Since, $p\left(2\right)=0$

Hence, $(x-2)$ is a factor of polynomial $3x^2+6x-24$.