Math
Zeros and multiplicity.
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
Real-world.
- 1 f(x) = (x − 2)³ has a zero at x = 2 with multiplicity 3 — the graph crosses and flattens at that point
- 2 f(x) = (x + 1)² has a zero at x = −1 with multiplicity 2 — the graph bounces off the x-axis
- 3 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