Section 5.1: Polynomial Functions – Definitions, Properties, and Identification

Polynomial Functions

Definition and Structure of Polynomial Functions

Polynomial functions are a fundamental class of functions in algebra and precalculus, characterized by their specific algebraic form and properties. Understanding their structure is essential for analyzing their graphs and behavior.

• Polynomial Function: A function in one variable of the form: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants (called coefficients), $n \geq 0$ is an integer (the degree), and $x$ is the variable.

• Leading Term: The term with the highest power of $x$ ($a_n x^n$ if $a_n \neq 0$).

• Constant Term: The term $a_0$, which does not contain $x$.

• Zero Polynomial: If all coefficients are zero, the function is called the zero polynomial, which has no degree.

• Standard Form: Polynomials are usually written in descending order of degree, starting with the leading term.

Domain: The domain of a polynomial function is the set of all real numbers.

Identifying Polynomial Functions and Their Degree

To determine whether a function is a polynomial, check that all exponents of $x$ are nonnegative integers and that the function is a sum of terms of the form $a_k x^k$.

• Example 1: $f(x) = 4x^3 - \frac{1}{2}x^2 - 7$ - This is a polynomial function of degree 3. - Leading term: $4x^3$ - Constant term: $-7$

• Example 2: $f(x) = x + 3 - 5x^4$ - Standard form: $-5x^4 + x + 3$ - Degree: 4 - Leading term: $-5x^4$ - Constant term: $3$

• Example 3: $g(x) = \sqrt[3]{x}$ - Not a polynomial function because the exponent ($1/3$) is not a nonnegative integer.

Key Properties of Polynomial Functions

• All exponents of $x$ must be whole numbers (nonnegative integers).

• Coefficients can be any real numbers.

• The graph of a polynomial function is always continuous (no gaps or holes) and smooth (no sharp corners).

Summary Table: Polynomial Function Terms

Degree ($n$)	General Form	Name	Graph Properties
0	$f(x) = a_0$	Constant function	Horizontal line, $y$-intercept at $a_0$
1	$f(x) = a_1 x + a_0$	Linear function	Nonvertical straight line, slope $a_1$
2	$f(x) = a_2 x^2 + a_1 x + a_0$	Quadratic function	Parabola, opens up if $a_2 > 0$, down if $a_2 < 0$

Additional info:

• Higher-degree polynomials ($n \geq 3$) have more complex graphs, with up to $n-1$ turning points.

• Polynomial functions are foundational for calculus and further mathematical analysis.