Example: Identifying Polynomial Functions

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]

Answer: 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.

Example: Identifying the Degree and Leading Coefficient of a Polynomial Function

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

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

Answer: For the function [latex]f\left(x\right)[/latex], the highest power of x is 3, so the degree is 3. 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 t is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[/latex]. The leading coefficient is the coefficient of that term, 5. For the function [latex]h\left(p\right)[/latex], the highest power of p is 3, so the degree is 3. 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].

Example: Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in the graph below.

Answer: As the input values x get very large, the output values [latex]f\left(x\right)[/latex] increase without bound. As the input values x get very small, the output values [latex]f\left(x\right)[/latex] decrease without bound. We can describe the end behavior symbolically by writing

[latex]\begin{array}{c}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to \infty \end{array}[/latex]

In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity. We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Example: Identifying End Behavior and Degree of a Polynomial Function

Given the function [latex]f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)[/latex], express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Answer: Obtain the general form by expanding the given expression for [latex]f\left(x\right)[/latex].

[latex]\begin{array}{c} f\left(x\right)=-3{x}^{2}\left(x - 1\right)\left(x+4\right)\\ \hfill =-3{x}^{2}\left({x}^{2}+3x - 4\right)\\ \hfill=-3{x}^{4}-9{x}^{3}+12{x}^{2}\end{array}[/latex]

The general form is [latex]f\left(x\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[/latex]. The leading term is [latex]-3{x}^{4}[/latex]; therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

[latex]\begin{array}{c}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left(x\right)\to -\infty \end{array}[/latex]