Chapter 4, Problem 2RE

In Problems $1-4$, determine whether the function is a polynomial function, a rational function, or neither. For those that are polynomial functions, state the degree. For those that are not polynomial functions, tell why not.

$f\left(x\right)=\frac{3x^5}{2x+1}$

To determine

Whether the function $f(x)=\frac{3x^5}{2x+1}$ is a polynomial function, a rational function, or neither. If polynomial, state degree, otherwise tell why not.

Answer to Problem 2RE

Solution:

The function $f(x)=\frac{3x^5}{2x+1}$ is a rational function and not polynomial function since it is ratio of two polynomials.

Explanation of Solution

Given Information:

The function, $f(x)=\frac{3x^5}{2x+1}$

The rational function is of the form $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ where $p$ and $q$ are polynomial functions and $q$ is not zero polynomial.

In a function $3x^5$ and $2x+1$ are polynomial functions and $2x+1$ is not the zero polynomial.

Hence, the function $f(x)=\frac{3x^5}{2x+1}$ is a rational function.

It is not polynomial function since it is ratio of two polynomials.