Lesson 4.4

Real Zeros and Multiplicity

Zeros tell us where the graph hits the x-axis. Multiplicity tells us how it hits it.

Introduction

Prerequisite Connection: We know solving gives the x-intercepts. But sometimes we get the same answer twice, like in .

Today's Increment: We learn that repeating roots (Multiplicity) change the graph's behavior. Odd repeated roots cross, but flatten out. Even repeated roots don't cross at all—they bounce.

Why This Matters for Calculus: Curve sketching is a lost art, but vital for visualizing functions without a calculator. Knowing multiplicity helps you predict local maximums and points of inflection (Calculus I).

Explanation of Key Concepts

Behavior at Intercepts

Multiplicity 1

(x-a)

Straight Cross

Even (2, 4...)

(x-a)²

bounce

Touch and Turn

Odd > 1 (3, 5...)

(x-a)³

∼

Flatten and Cross

Worked Examples

Level: Basic

Example 1: Identifying Behaviors

Find zeros and behavior for .

x = -2
Exponent is 1 (Odd).
Standard Cross.
x = 1
Exponent is 2 (Even).
Touch/Bounce.
x = 4
Exponent is 3 (Odd > 1).
Flatten/Wiggle Cross.

Level: Intermediate

Example 2: The "Bounce" Effect

Graph .

Notice at -1, it acts like a parabola (bounces back down). At 2, it acts like a line (slices through).

Level: Advanced

Example 3: Building Equation from Graph

A degree 4 polynomial bounces at and crosses at and . Write a possible equation.

Step 1: Assign Factors

Root 3 (Bounce) → (Must be even)
Root -1 (Cross) →
Root 5 (Cross) →

Step 2: Check Degree

. This matches the requirement!

Answer

for any

Common Pitfalls

Mixing up x and the root:
If the factor is , the root is NOT 5. It is .
Forgetting the "Flatten" behavior:
Students often draw as a straight line crossing the axis. Remember to make it "wiggle" or flatten out momentarily at the intercept to show the triple root.

Real-World Application

Structural Engineering: Deflection

When calculating how a beam bends under load, the boundary conditions are essentially roots.

A "simply supported" beam (resting on pins) acts like a single root (can rotate but not move). A "cantilevered" beam (embedded in a wall) acts like a double root or higher because the slope is also constrained to be zero. The multiplicity of the root describes the physical constraint!