Question

Use the factor theorem to determine whether $(x-1)$ is a factor of:

(iii) $2\sqrt{2}{x}^3+5\sqrt{2}{x}^2-7\sqrt{2}$

Answer 1

Checking for the factor.
Factor theorem: Let $g\left(x\right)$ be a polynomial of degree greater than or equal to 1 and $a$ be a real number. Then the following holds:

$g\left(a\right)=0\Rightarrow (x-a)$ is a factor of $g\left(x\right)$

$(x-a)$ is a factor of $g\left(x\right)$ $\Rightarrow$$g\left(a\right)=0$.

Let $g\left(x\right)=2\sqrt{2}{x}^3+5\sqrt{2}{x}^2-7\sqrt{2}$

According to the factor theorem, if $(x-1)$ is a factor of $g\left(x\right)$ then $g\left(1\right)=0$.

Substitute 1 for $x$ in the function $g\left(x\right)$ to obtain $g\left(1\right)$

$$\begin{gathered} g\left(1\right)=2\sqrt{2}{\left(1\right)}^3+5\sqrt{2}{\left(1\right)}^2-7\sqrt{2} \\ \Rightarrow g\left(1\right)=2\sqrt{2}+5\sqrt{2}-7\sqrt{2} \\ \therefore g\left(1\right)=0 \end{gathered}$$

As $g\left(1\right)=0$, $(x-1)$ is a factor of $g\left(x\right)$.

Hence, $(x-1)$ is a factor of the given polynomial.