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?

[latex]\begin{array}{ccc}f\left(x\right)=5x^7+4\hfill \\ g\left(x\right)=-x^2\left(x-\dfrac{2}{5}\right)\hfill \\ h\left(x\right)=\dfrac{1}{2}x^2+\sqrt{x}+2\hfill \end{array}[/latex]

In the following video, you will see additional examples of how to identify a polynomial function using the definition.

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]

Watch the next video for more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.

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:

  1. Find the highest power of x to determine the degree of the function.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.