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.
[latex]\left(w\right)=24+8w[/latex]
We can combine this with the formula for the area A of a circle.
[latex]\left(w\right)=\pi {r}^{2}[/latex]
Composing these functions gives a formula for the area in terms of weeks.
[latex]\begin{cases}\left(w\right)=\left(\left(\right)\right)\\ =\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{cases}[/latex]
Multiplying gives the formula.
[latex]\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}[/latex]
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
[latex]f\left(\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
This is called the general form of a polynomial function. Each [latex]{a}_{i}[/latex] is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[/latex] is a term of a polynomial function.
Example 4: Identifying Polynomial Functions
Which of the following are polynomial functions?
[latex]\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}[/latex]
Solution
The first two functions are examples of polynomial functions because they can be written in the form [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex], where the powers are non-negative integers and the coefficients are real numbers.
- [latex]f\left(x\right)[/latex]
can be written as [latex]f\left(x\right)=6{x}^{4}+4[/latex]. - [latex]g\left(x\right)[/latex]
can be written as [latex]g\left(x\right)=-{x}^{3}+4x[/latex]. - [latex]h\left(x\right)[/latex]
cannot be written in this form and is therefore not a polynomial function.