Basic Characteristics of Polynomial Functions
Learning Outcomes
- Determine if a given function is a polynomial function
- Determine the degree and leading coefficient of a polynomial function
Recognize Polynomial Functions
We have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions. Polynomials are algebraic expressions that are created by adding or subtracting monomial terms, such as [latex]-3x^2[/latex], where the exponents are only non-negative integers. Functions are a specific type of relation in which each input value has one and only one output value. Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section, we will identify and evaluate polynomial functions. Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the powers of the variables. When we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[/latex]. We can turn this into a polynomial function by using function notation:[latex]f(x)=4x^3-9x^2+6x[/latex]
Polynomial functions are written with the leading term first and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.
Example
Which of the following are polynomial functions?
Answer:
The first two functions are examples of polynomial functions because they contain powers that are non-negative integers and the coefficients are real numbers. Note that the second function can be written as [latex]g\left(x\right)=-x^3+\dfrac{2}{5}x[/latex] after applying the distributive property.
The third function is not a polynomial function because the variable is under a square root in the middle term, therefore the function contains an exponent that is not a non-negative integer.Determine the Degree and Leading Coefficient of a Polynomial Function
Just as we identified the degree of a polynomial, we can identify the degree of a polynomial function. To review: the degree of the polynomial is the highest power of the variable that occurs in the polynomial; the leading term is the term containing the highest power of the variable or the term with the highest degree. The leading coefficient is the coefficient of the leading term.Example
Identify the degree, leading term, and leading coefficient of the following polynomial functions.[latex]\begin{array}{lll} f\left(x\right)=5{x}^{2}+7-4{x}^{3} \\ g\left(x\right)=9x-{x}^{6}-3{x}^{4}\\ h\left(x\right)=6\left(x^2-x\right)+11\end{array}[/latex]
Answer:
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(x\right)[/latex], the highest power of [latex]x[/latex] is [latex]6[/latex], so the degree is [latex]6[/latex]. The leading term is the term containing that degree, [latex]-{x}^{6}[/latex]. The leading coefficient is the coefficient of that term, [latex]-1[/latex].
For the function [latex]h\left(x\right)[/latex], first rewrite the polynomial using the distributive property to identify the terms. [latex]h\left(x\right)=6x^2-6x+11[/latex]. The highest power of [latex]x[/latex] is [latex]2[/latex], so the degree is [latex]2[/latex]. The leading term is the term containing that degree, [latex]6{x}^{2}[/latex]. The leading coefficient is the coefficient of that term, [latex]6[/latex].
Summary
Polynomial functions contain powers that are non-negative integers and the coefficients are real numbers. It is often helpful to know how to identify the degree and leading coefficient of a polynomial function. To do this, follow these suggestions:- Find the highest power of x to determine the degree of the function.
- Identify the term containing the highest power of x to find the leading term.
- Identify the coefficient of the leading term.
Licenses & Attributions
CC licensed content, Original
- Determine if a Function is a Polynomial Function. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Degree, Leading Term, and Leading Coefficient of a Polynomial Function. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: Open Stax Authored by: Abramson, Jay, et al. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download fro free at : http://cnx.org/contents/[email protected]:1/Preface.
