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

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

Past Knowledge

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

Today's Goal

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.

Future Success

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).

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-Life Applications

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!