Question

Use the Remainder Theorem to find the remainder when $f\left(x\right)$is divided by $x-c$. Then use the Factor Theorem to determine whether $x-c$is a factor of $f\left(x\right)$.

$f\left(x\right)=-4x^3+5x^2+8;x+3$

Short answer

The remainder is$161$and$x+3$is not a factor.

Step-by-step solution

Step 1. Given information.

The given function is,

$f\left(x\right)=-4x^3+5x^2+8;x+3$.

Step 2. Using remainder theorem.

We know that if $f\left(x\right)$is divided by $x-c$then the remainder will be $f\left(c\right)$.

Here $f\left(x\right)$is divided by $x+3$, then the remainder is $f(-3)$that is

$$\begin{gathered} f(-3)=-4{\left(-3\right)}^3+5{\left(-3\right)}^2+8 \\ =-4\left(-27\right)+5\left(9\right)+8 \\ =108+45+8 \\ =161 \end{gathered}$$

Therefore, the remainder is $161$.

Step 3. Using factor theorem.

We know that if $x-c$is the factor of the polynomial $f\left(x\right)$then

$f\left(c\right)=0$,

Here $c=-3$, then

$$\begin{gathered} f\left(c\right)=f(-3) \\ =161 \end{gathered}$$

We observe that $f(-3)\ne 0$,then $x+3$is not a factor.