Question

Use factor theorem to find if $x+\sqrt{2}$ is a factor of $2\sqrt{2}{x}^2+5x+\sqrt{2}$.

Answer 1

Let $f\left(x\right)=2\sqrt{2}{x}^2+5x+\sqrt{2}$.

If $(x+\sqrt{2})$ is a factor of $f\left(x\right)$, then $f(-\sqrt{2})=0$.

Now, $f(-\sqrt{2})=\left(2\sqrt{2}\right).2-5\left(\sqrt{2}\right)+\sqrt{2}$

$=5\sqrt{2}-5\sqrt{2}=0$

$\therefore (x+\sqrt{2})$ is a factor of $f\left(x\right)$.