## Identify polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear.
$\left(w\right)=24+8w$
We can combine this with the formula for the area A of a circle.
$\left(w\right)=\pi {r}^{2}$
Composing these functions gives a formula for the area in terms of weeks.
$\begin{cases}\left(w\right)=\left(\left(\right)\right)\\ =\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{cases}$
Multiplying gives the formula.
$\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}$
This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

### A General Note: Polynomial Functions

Let n be a non-negative integer. A polynomial function is a function that can be written in the form
$f\left(\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$
This is called the general form of a polynomial function. Each ${a}_{i}$ is a coefficient and can be any real number. Each product ${a}_{i}{x}^{i}$ is a term of a polynomial function.

### Example 4: Identifying Polynomial Functions

Which of the following are polynomial functions?
$\begin{cases}f\left(x\right)=2{x}^{3}\cdot 3x+4\hfill \\g\left(x\right)=-x\left({x}^{2}-4\right)\hfill \\ h\left(x\right)=5\sqrt{x}+2\hfill \end{cases}$

### Solution

The first two functions are examples of polynomial functions because they can be written in the form $f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$, where the powers are non-negative integers and the coefficients are real numbers.

• $f\left(x\right)$
can be written as $f\left(x\right)=6{x}^{4}+4$.
• $g\left(x\right)$
can be written as $g\left(x\right)=-{x}^{3}+4x$.
• $h\left(x\right)$
cannot be written in this form and is therefore not a polynomial function.