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.

[latex]rleft(wright)=24+8w\[/latex]

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

[latex]Aleft(rright)=pi {r}^{2}\[/latex]

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

[latex]begin{cases}Aleft(wright)=Aleft(rleft(wright)right)\ =Aleft(24+8wright)\ =pi {left(24+8wright)}^{2}end{cases}\[/latex]

Multiplying gives the formula.

[latex]Aleft(wright)=576pi +384pi w+64pi {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]fleft(xright)={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}fleft(xright)=2{x}^{3}cdot 3x+4hfill \ gleft(xright)=-xleft({x}^{2}-4right)hfill \ hleft(xright)=5sqrt{x}+2hfill end{cases}\[/latex]

Solution

The first two functions are examples of polynomial functions because they can be written in the form [latex]fleft(xright)={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]fleft(xright)\[/latex] can be written as [latex]fleft(xright)=6{x}^{4}+4\[/latex].
[latex]gleft(xright)\[/latex] can be written as [latex]gleft(xright)=-{x}^{3}+4x\[/latex].
[latex]hleft(xright)\[/latex] cannot be written in this form and is therefore not a polynomial function.

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Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..

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