Zeros and multiplicity

Math

Definition

The multiplicity of a zero is the number of times its corresponding factor appears in the factored form of a polynomial. Multiplicity affects how the graph behaves at that zero: odd multiplicity means the graph crosses the x-axis, while even multiplicity means it touches and bounces back.

Examples

  • f(x) = (x − 2)³ has a zero at x = 2 with multiplicity 3 — the graph crosses and flattens at that point
  • f(x) = (x + 1)² has a zero at x = −1 with multiplicity 2 — the graph bounces off the x-axis
  • f(x) = x(x − 3)²(x + 2) has zeros at 0 (mult. 1), 3 (mult. 2), and −2 (mult. 1)
Key Fact

Odd multiplicity → graph crosses x-axis; Even multiplicity → graph touches and bounces

Related Terms

Zeros and end behavior Solving by factoring Fundamental theorem of algebra Polynomial operations

Study This Concept

Practice zeros and multiplicity with free review games in these units:

Pre-Calculus Unit 2: Polynomial and Rational Functions