5.4.1: Resources and Key Concepts

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Resources

Videos

Graphs of polynomial functions
- Graphs of Polynomial Functions - Graphing Polynomial Functions
- Polynomial Functions and Their Graphs - Graphing Polynomials Using End Behavior and Zeros
Zeros (and their multiplicities) of polynomial functions
- Graphs of Polynomial Functions - Identifying Zeros and Their Multiplicities
- Polynomial Functions and Their Graphs - Zeros of Functions and Their Multiplicities
Factoring polynomials
- Graphs of Polynomial Functions - Using Factoring to Find Zeros of Polynomial Functions
Polynomial modeling
- Polynomial Functions - Applications

Key Concepts

Definitions

Zeros of a Polynomial Function: The values of \(x\) for which \(f(x)=0\). These correspond to the \(x\)-intercepts of the graph.
Multiplicity of a Zero: If a polynomial contains a factor of the form \((x-h)^p\), and if this is the greatest such factor of \((x-p)\), we say that \(x=h\) is a zero of multiplicity \(p\).
Single Zero: A zero of multiplicity 1. The graph passes directly through the \(x\)-axis at a single zero.

Theorems

Local Behavior Near a Zero: If \(x=h\) is a zero of multiplicity \(p\) for a given polynomial, then the local behavior of the graph of the polynomial near \(x=h\) matches the behavior of \(y = \pm (x-h)^p\), where the sign of the coefficient is determined by other characteristics of the given polynomial.
Sum of Multiplicities and the Degree of a Polynomial: The sum of the multiplicities of the factors for a polynomial is the degree of the polynomial function.
Intermediate Value Theorem (for Polynomials): Let \(f\) be a polynomial function. If \(f(a)\) and \(f(b)\) have opposite signs, then there exists at least one value \(c\) between \(a\) and \(b\) for which \(f(c)=0\).

Common Mistakes

Incorrectly Identifying Multiplicity: Miscounting the multiplicity of a zero, especially when a factor is repeated or when a zero comes from a factor like \((x^2-k)\) which might yield two distinct zeros each of multiplicity 1.
Misinterpreting Behavior at Zeros based on Multiplicity: Confusing whether the graph crosses or touches/bounces at an \(x\)-intercept based on the multiplicity of the zero (odd multiplicity means cross, even multiplicity means touch/bounce).
Errors in Determining End Behavior: Incorrectly identifying the leading term (especially if the polynomial is not in standard form) or misapplying the rules for end behavior based on the degree and sign of the leading coefficient.
Exceeding Maximum Number of Turning Points: Sketching a graph with more turning points than \(n-1\), where \(n\) is the degree of the polynomial.

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