Question

In Problems 13, find the remainder R when $f\left(x\right)$is divided by $g\left(x\right)$. Is $g$a factor of $f$?

$f\left(x\right)=x^4-2x^3+15x-2;g\left(x\right)=x+2$

Short answer

The remainder R is $0$.

Yes,$g\left(x\right)$is a factor of$f\left(x\right)$

Step-by-step solution

Given information

$f\left(x\right)=x^4-2x^3+15x-2;g\left(x\right)=x+2$

Determine whether g is a factor of f

Note: When $f\left(x\right)$is divided by a linear polynomial $g\left(x\right)$, the remainder $R$is equal to $f\left(c\right)$based from the Remainder Theorem.

$c$is a real number from $g\left(x\right)=x-c$. In this problem, $c=-2$.

Substitute $-2$for $x$in $f\left(x\right)$.

$\begin{array}{rcl}f\left(-2\right)& =& {\left(-2\right)}^4-2{\left(-2\right)}^3+15\left(-2\right)-2\\ & =& 16+16-30-2\\ & =& 0\end{array}$

Since $R=0,g\left(x\right)$ is a factor of $f\left(x\right)$.

Write the conclusion

The remainder is$0$.

Yes, $g\left(x\right)$is a factor of$f\left(x\right)$.