Lesson 2.16

Multiplicity of Zeros

Some zeros repeat — and the number of times they repeat tells you how the graph behaves at the x-axis: does it cross through, or bounce off?

Introduction

When appears as a factor more than once, the zero has a multiplicity greater than 1. This multiplicity controls whether the graph crosses, bounces, or flattens at that intercept.

Past Knowledge

Factoring polynomials completely (Chapters 7-8).

Today's Goal

Determine multiplicity from factored form and predict graph behavior at each zero.

Future Success

Multiplicity is the key to sketching accurate polynomial graphs in Lesson 2.18.

Key Concepts

What Is Multiplicity?

If is a factor of , then is a zero with multiplicity .

Example:

→ multiplicity 3

→ multiplicity 2

Bounce vs. Cross

Odd Multiplicity → Crosses

The graph passes through the x-axis. Multiplicity 1 looks like a line; multiplicity 3 flattens first, then crosses (an "S" shape).

Even Multiplicity → Bounces

The graph touches the x-axis and turns around, like a parabola at its vertex. The higher the multiplicity, the flatter the bounce.

Quick Reference

Multiplicity	Behavior	Visual
1 (odd)	Crosses straight	╱
2 (even)	Bounces (U-turn)	∪
3 (odd)	Flat cross (S-shape)	∿

Worked Examples

Example 1: Identify Multiplicities

Basic

State the zeros and their multiplicities for .

Zero	Factor	Multiplicity	Behavior
		2 (even)	Bounces
		1 (odd)	Crosses
		3 (odd)	Flat cross

Degree check:

— the polynomial is degree 6 ✓

Example 2: Factor First, Then Analyze

Intermediate

Find zeros and their multiplicities: .

Factor out the GCF

Factor completely

Analyze

with multiplicity 2 → Bounces

Both intercepts are tangent to the x-axis — the graph never crosses!

Example 3: Build From Zeros

Advanced

Write a polynomial of lowest degree that bounces at , crosses at , and has a leading coefficient of 1.

Assign Multiplicities

Bounce → even multiplicity (minimum 2). Cross → odd multiplicity (minimum 1).

Write in Factored Form

Expand (Optional Verification)

Degree 3 = minimum degree (2 + 1) ✓

Common Pitfalls

Confusing Bounce and Cross

Remember: even multiplicity = bounce. Odd multiplicity = cross. A common mnemonic: an even number "evens out" — the graph comes back the way it came.

Ignoring the GCF

When is factored out, that gives with multiplicity 2. Students often lose the zero at the origin when they factor out but forget to record it.

Real-Life Applications

In mechanical engineering, multiplicity appears in vibration analysis. A repeated root in the characteristic equation of a vibrating system means the system is at a critical damping point — gravity between an oscillating and a non-oscillating response. This is how suspension systems are tuned.