Question

True or False If $f$is a polynomial function of degree $4$and if $f\left(2\right)=5$, then

role="math" localid="1646217783867" $\frac{f\left(x\right)}{x-2}=p\left(x\right)+\frac{5}{x-2}$

where $p\left(x\right)$ is a polynomial function of degree $3$.

Short answer

The given statement is true.

Step-by-step solution

Step 1. GIven information.

The given statement is,

If $f$is a polynomial function of degree $4$and if $f\left(2\right)=5$, then

$\frac{f\left(x\right)}{x-2}=p\left(x\right)+\frac{5}{x-2}$

where $p\left(x\right)$is a polynomial function of degree $3$.

Step 2. Explanation.

We know that if $f$is a polynomial divided by $x-c$then the degree of the quotient polynomial is one less than that of the polynomial $f$.

Here the degree of polynomial $f$is $4$and divided by $x-2$then the degree of the quotient $p\left(x\right)$will be $3$.

Therefore, the given statement is true.