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 $f$?

$f\left(x\right)=8x^3-3x^2+x+4;g\left(x\right)=x-1$

Short answer

The remainder R is $10$.

No,$g\left(x\right)$is not a factor of$f\left(x\right)$

Step-by-step solution

Given information

The given function is$f\left(x\right)=8x^3-3x^2+x+4;g\left(x\right)=x-1$

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, $c=1$.

Substitute $1$for $x$in $f\left(x\right)$.

$\begin{array}{rcl}f\left(x\right)& =& 8{\left(1\right)}^3-3{\left(1\right)}^2+\left(1\right)+4\\ & =& 8-3+1+4\\ & =& 10\end{array}$

Since,$R=10$,$g\left(x\right)$is not a factor of$f\left(x\right)$

Write the conclusion

The remainder is $10$.

No,$g\left(x\right)$is not a factor of$f\left(x\right)$.