Back4.1 Polynomial Functions: Definitions, Properties, and Examples
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Polynomial Functions
Definition and General Form
A polynomial function in one variable is a function of the form:
Coefficients: are constants and real numbers.
Degree: is a non-negative integer (i.e., ) and is called the degree of the polynomial.
Leading Coefficient: is the coefficient of the highest power of ; if , it is called the leading coefficient.
Variable: is the variable of the function.
The domain of a polynomial function is the set of all real numbers, .
Properties and Types of Polynomial Functions
Polynomial functions can be classified by their degree. The following table summarizes the main types:
Degree | Name | Graph |
|---|---|---|
No degree | Zero function | The x-axis |
0 | Constant function | Horizontal line with y-intercept |
1 | Linear function | Nonvertical, nonhorizontal line with slope and y-intercept |
2 | Quadratic function | Parabola: graph is concave up if , concave down if |
Key Terms
Zero Function: for all .
Constant Function: where is a real number.
Linear Function: where .
Quadratic Function: where .
Examples and Applications
Determine if the following are polynomial functions. If so, state their degrees:
Answer: Yes, this is a constant polynomial function. Degree: 0.
Answer: Yes, this is a quadratic polynomial function. Degree: 2.
Answer: Yes, this is a polynomial function. Expand to find the degree: , so the highest degree is 9. Degree: 9.
Answer: No, this is not a polynomial function because it contains an exponential term.
Answer: No, this is not a polynomial function because the exponent is not a non-negative integer.
Which functions (if any) from Question 1 are linear?
From the list above, only is a constant function (degree 0), and is quadratic (degree 2). None of the listed functions are linear (degree 1).
Additional info:
Polynomial functions are fundamental in algebra and calculus, forming the basis for more advanced topics such as polynomial equations, factoring, and graphing.
Recognizing the degree and leading coefficient helps predict the end behavior and shape of the graph.
