Algebra 2 · Unit 3 · Topic 3.5
Zeros, Multiplicity, and Polynomial Graphs
A zero's multiplicity is the exponent of its factor in a polynomial's complete factorization.
Definition and explanation
A zero's multiplicity is the exponent of its factor in a polynomial's complete factorization. A real zero of odd multiplicity changes the polynomial's sign; one of even multiplicity does not, so the graph touches the axis rather than crossing there.
Worked example
For p(x) = (x + 1)²(x − 3), x = −1 has multiplicity 2 and x = 3 has multiplicity 1. The graph touches at −1 and crosses at 3. Also, p(0) = −3 locates its y-intercept.
Common mistake
The number of distinct zeros can be smaller than the degree because zeros may repeat or be nonreal.
Key takeaway
A zero's multiplicity is the exponent of its factor in a polynomial's complete factorization.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.