• Recognize polynomial functions, noting their degree and leading coefficient
• Apply various methods to divide polynomials and locate the zeros of polynomial equations
• Generate graphs and formulate equations for polynomial functions
• Explain the end behavior of polynomial functions

Identifying Polynomial Functions

An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[/latex] miles in radius, but that radius is increasing by [latex]8[/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[/latex] of the spill depends on the number of weeks [latex]w[/latex] that have passed. This relationship is linear.

[latex]r\left(w\right)=24+8w[/latex]

We can combine this with the formula for the area [latex]A[/latex] of a circle.

[latex]A\left(r\right)=\pi {r}^{2}[/latex]

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

[latex]\begin{array}{l}A\left(w\right)=A\left(r\left(w\right)\right)\\ A\left(w\right)=A\left(24+8w\right)\\ A\left(w\right)=\pi {\left(24+8w\right)}^{2}\end{array}[/latex]

Multiplying gives the formula below.

[latex]A\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.

polynomial functions

Let [latex]n[/latex]  be a non-negative integer. A polynomial function is a function that 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]

 

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.

Which of the following are polynomial functions?

[latex]\begin{array}{c}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{array}[/latex]

Show 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.

Defining the Degree and Leading Coefficient of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form. 

terminology of polynomial functions

 

The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.

 

The leading term is the term containing the variable with the highest power, also called the term with the highest degree.

 

The leading coefficient is the coefficient of the leading term.

How To: Given a Polynomial Function, Identify the Degree and Leading Coefficient

1. Find the highest power of [latex]x[/latex] to determine the degree of the function.
2. Identify the term containing the highest power of [latex]x[/latex] to find the leading term.
3. The leading coefficient is the coefficient of the leading term.

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

[latex]\begin{array}{l} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\h\left(p\right)=6p-{p}^{3}-2\end{array}[/latex]

Show Solution

For the function [latex]f\left(x\right)[/latex], the highest power of [latex]x[/latex] is [latex]3[/latex], so the degree is [latex]3[/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[/latex]. The leading coefficient is the coefficient of that term, [latex]–4[/latex].

For the function [latex]g\left(t\right)[/latex], the highest power of [latex]t[/latex] is [latex]5[/latex], so the degree is [latex]5[/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[/latex]. The leading coefficient is the coefficient of that term, [latex]5[/latex].

For the function [latex]h\left(p\right)[/latex], the highest power of [latex]p[/latex] is [latex]3[/latex], so the degree is [latex]3[/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[/latex]; the leading coefficient is the coefficient of that term, [latex]–1[/latex].