Polynomial Functions: Structure, Graphs, Zeros, and Turning Points

Polynomial Functions

Introduction to Polynomial Functions

Polynomial functions are a fundamental class of functions in algebra, characterized by expressions involving only non-negative integer powers of the variable. Understanding their structure and graphical behavior is essential for analyzing more complex mathematical models.

• Definition: A polynomial function is a function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a non-negative integer and the coefficients $a_i$ are real numbers.

• Standard Form: Terms are written in descending order of power.

• Degree: The highest exponent of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term without $x$.

Example:

• $f(x) = 6x^3 + 3x^2 + 5x + 4$ Degree: 3 Leading Coefficient: 6 Constant: 4

Identifying Polynomial Functions

To determine if a function is a polynomial, check that all exponents are non-negative integers and coefficients are real numbers.

• Polynomial: $f(x) = -x^4 + 5x^2 - 6x + 4$ (Degree: 4, Leading Coefficient: -1)

• Not a Polynomial: $f(x) = 2x^{1/3}$ (Exponent is not a whole number)

Graphs of Polynomial Functions

The graphs of polynomial functions are always smooth and continuous, meaning they have no sharp corners or breaks.

• Polynomial Graphs: Smooth curves, no gaps or jumps.

• Non-Polynomial Graphs: May have sharp points, discontinuities, or breaks.

End Behavior of Polynomial Functions

Understanding End Behavior

The end behavior of a polynomial function describes how the function behaves as $x$ approaches $+\infty$ or $-\infty$. This is determined by the degree and leading coefficient.

• If the degree is even and the leading coefficient is positive, both ends rise.

• If the degree is even and the leading coefficient is negative, both ends fall.

• If the degree is odd and the leading coefficient is positive, left end falls, right end rises.

• If the degree is odd and the leading coefficient is negative, left end rises, right end falls.

Example:

• $f(x) = 3x^5 + \ldots$ Degree: 5 (odd), Leading Coefficient: 3 (positive) Left end falls, right end rises.

Zeros and Multiplicity

Finding Zeros by Factoring

The zeros (or roots) of a polynomial are the values of $x$ for which $f(x) = 0$. These can be found by factoring the polynomial and setting each factor equal to zero.

• Multiplicity: The number of times a particular zero occurs (i.e., the exponent of the factor).

• If the zero has even multiplicity, the graph touches the x-axis at that point.

• If the zero has odd multiplicity, the graph crosses the x-axis at that point.

Example:

• $f(x) = 2x(x-3)^2(x+4)^3$ Zeros: $x=0$ (multiplicity 1, crosses), $x=3$ (multiplicity 2, touches), $x=-4$ (multiplicity 3, crosses)

Turning Points of Polynomial Functions

Maximum Number of Turning Points

A turning point is a point where the graph changes direction from increasing to decreasing or vice versa. The maximum number of turning points for a polynomial of degree $n$ is $n-1$.

• Local Maximum: Highest point in a local region.

• Local Minimum: Lowest point in a local region.

• Maximum Turning Points: $n-1$ for degree $n$.

Example:

• $f(x) = 6x^4 + 2x$ Degree: 4 Maximum turning points: 3

Summary Table: End Behavior Based on Degree and Leading Coefficient

Degree	Leading Coefficient	Left End	Right End
Even	Positive	Rises	Rises
Even	Negative	Falls	Falls
Odd	Positive	Falls	Rises
Odd	Negative	Rises	Falls

Practice Problems

• Determine if $f(x) = 4x^2 + \frac{1}{2}x^2 - 2x + 1$ is a polynomial. If so, state its degree and leading coefficient. Answer: Yes, degree 2, leading coefficient 4.5

• Find the zeros and their multiplicities for $f(x) = x^2(x-1)^2(x+6)$. Answer: $x=0$ (multiplicity 2, touches), $x=1$ (multiplicity 2, touches), $x=-6$ (multiplicity 1, crosses)

• Determine the maximum number of turning points for $f(x) = 3x^4 + 2x$. Answer: 3

Additional info: These notes cover the foundational aspects of polynomial functions, including their structure, graphical properties, zeros, multiplicity, and turning points, as outlined in College Algebra Chapter 4.