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Precalculus
17. CC.HSA.APR.C.4
17.2 Multiplicities of polynomials
Multiplicity of Zeros in Polynomials
See how the multiplicity of a zero tells you whether a polynomial graph crosses or bounces off the x-axis at each root.
What You'll Learn
The multiplicity of a zero is the number of times its factor appears when the polynomial is fully factored.
A zero with odd multiplicity makes the graph cross straight through the x-axis at that point.
A zero with even multiplicity makes the graph touch the x-axis and bounce back without crossing.
The sum of all multiplicities always equals the degree of the polynomial.
Higher odd multiplicities (3, 5, ...) make the graph flatten out as it crosses the axis.
What You'll Practice
1
Analyzing graphs to determine multiplicities of zeros
2
Sketching polynomial graphs using x-intercepts, y-intercepts, and multiplicities
3
Finding missing multiplicities when given polynomial degree
4
Identifying bounce points and crossing points from multiplicity values
Why This Matters
Understanding multiplicities is essential for graphing polynomials accurately and analyzing their behavior. This skill connects directly to calculus concepts like critical points and is fundamental for advanced algebra, helping you predict how functions behave without relying on a calculator.
This Unit Includes
3 Video lessons
Learning resources
Skills
Multiplicity
Polynomials
Graphing
X-intercepts
Zeros
Concavity
Even and Odd
IL Curriculum Aligned