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Study Guides > College Algebra

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.

r(w)=24+8wr\left(w\right)=24+8w\\

We can combine this with the formula for the area A of a circle.

A(r)=πr2A\left(r\right)=\pi {r}^{2}\\

Composing these functions gives a formula for the area in terms of weeks.

{A(w)=A(r(w))=A(24+8w)=π(24+8w)2\begin{cases}A\left(w\right)=A\left(r\left(w\right)\right)\\ =A\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{cases}\\

Multiplying gives the formula.

A(w)=576π+384πw+64πw2A\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(x)=anxn+⋯+a2x2+a1x+a0f\left(x\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 ai{a}_{i}\\ is a coefficient and can be any real number. Each product aixi{a}_{i}{x}^{i}\\ is a term of a polynomial function.

Example 4: Identifying Polynomial Functions

Which of the following are polynomial functions?

{f(x)=2x3⋅3x+4g(x)=−x(x2−4)h(x)=5x+2\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(x)=anxn+⋯+a2x2+a1x+a0f\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(x)f\left(x\right)\\ can be written as f(x)=6x4+4f\left(x\right)=6{x}^{4}+4\\.
  • g(x)g\left(x\right)\\ can be written as g(x)=−x3+4xg\left(x\right)=-{x}^{3}+4x\\.
  • h(x)h\left(x\right)\\ cannot be written in this form and is therefore not a polynomial function.

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