Question

In the following case, use factor theorem to find whether $g\left(x\right)$ is a factor of the polynomial $p\left(x\right)$ or not.
$p\left(x\right)=2x^4+3x^3-2x^2-9x-12g\left(x\right)=x^2-3$

Answer 1

$g\left(x\right)=x^2-3$

$⟹g\left(x\right)=(x+\sqrt{3})(x-\sqrt{3})$

Now,

$p\left(x\right)=2x^4+3x^3-2x^2-9x-12$

$p\left(\sqrt{3}\right)=2(\sqrt{3}{)}^4+3(\sqrt{3}{)}^3-2(\sqrt{3}{)}^2-9(\sqrt{3})-12$

$=2\times 9+3\times 3\sqrt{3}-2\times 3-9\sqrt{3}-12$

$=18+9\sqrt{3}-18-9\sqrt{3}=0$

$\therefore (x-\sqrt{3})$ is a factor of $p\left(x\right)$

$p(-\sqrt{3})=2(-\sqrt{3}{)}^4+3(-\sqrt{3}{)}^3-2(-\sqrt{3}{)}^2-9(-\sqrt{3})-12$

$=2\times 9+3\times -3\sqrt{3}-2\times 3+9\sqrt{3}-12$

$=18-9\sqrt{3}-18+9\sqrt{3}=0$

$\therefore (x+\sqrt{3})$ is a factor $p\left(x\right)$

$\because$ both are the facors of $p\left(x\right),g\left(x\right)$ is a factor of the polynomial $p\left(x\right)$