Question

In each of the following, use factor theorem to find whether polynomial $g\left(x\right)$ is a factor of polynomial $f\left(x\right)$ or not:

$\left(2\right)$ $\begin{array}{c}f\left(x\right)=3x^4+17x^3+9x^2-\\ 7x-10;g\left(x\right)=x+5\end{array}$

Answer 1

Given; $f\left(x\right)=3x^4+17x^3+9x^2-7x-10;$

$g\left(x\right)=x+5$

$g\left(x\right)=x+5=x-(-5)$

If $f\left(x\right)$ is completely divided by $g\left(x\right)=(x-a)$ then by factor theorem $(x-a)$ is factor of $f\left(x\right)$ and $f\left(a\right)=0.$

$\because f(-5)=3\times (-5{)}^4+17\times (-5{)}^3+9\times (-5{)}^2$

$-7\times (-5)-10$

$=1875-2125+225+35-10$

$=2135-2135$

$=0$

$\because f(-5)=0$

Yes, $g\left(x\right)=(x+5)$ is a factor of $f\left(x\right)=3x^4+17x^3+9x^2-7x-10.$