4.1 Polynomial Functions: Definitions, Properties, and Examples

Polynomial Functions

Definition and General Form

A polynomial function in one variable is a function of the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• Coefficients: $a_n, a_{n-1}, \ldots, a_0$ are constants and real numbers.

• Degree: $n$ is a non-negative integer (i.e., $n = 0, 1, 2, \ldots$) and is called the degree of the polynomial.

• Leading Coefficient: $a_n$ is the coefficient of the highest power of $x$; if $a_n \neq 0$, it is called the leading coefficient.

• Variable: $x$ is the variable of the function.

The domain of a polynomial function is the set of all real numbers, $\mathbb{R}$.

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 $a_0$
1	Linear function	Nonvertical, nonhorizontal line with slope $a_1$ and y-intercept $a_0$
2	Quadratic function	Parabola: graph is concave up if $a_2 > 0$, concave down if $a_2 < 0$

Key Terms

• Zero Function: $f(x) = 0$ for all $x$.

• Constant Function: $f(x) = a_0$ where $a_0$ is a real number.

• Linear Function: $f(x) = a_1 x + a_0$ where $a_1 \neq 0$.

• Quadratic Function: $f(x) = a_2 x^2 + a_1 x + a_0$ where $a_2 \neq 0$.

Examples and Applications

Determine if the following are polynomial functions. If so, state their degrees:

1. $f(x) = 2$ Answer: Yes, this is a constant polynomial function. Degree: 0.

2. $h(x) = 6x^2 - 3x + 5$ Answer: Yes, this is a quadratic polynomial function. Degree: 2.

3. $g(x) = -5x^7(x - 4)^2$ Answer: Yes, this is a polynomial function. Expand to find the degree: $x^7(x-4)^2 = x^7(x^2 - 8x + 16) = x^9 - 8x^8 + 16x^7$, so the highest degree is 9. Degree: 9.

4. $f(x) = 4e^x$ Answer: No, this is not a polynomial function because it contains an exponential term.

5. $h(x) = 2 - (x - 1)^{1/3}$ Answer: No, this is not a polynomial function because the exponent $1/3$ is not a non-negative integer.

Which functions (if any) from Question 1 are linear?

• From the list above, only $f(x) = 2$ is a constant function (degree 0), and $h(x) = 6x^2 - 3x + 5$ 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.