Identify Zeros and Graph Polynomials

Lesson 3 of 7: Arithmetic with Polynomials

In this lesson:

• Read zeros directly from factored form
• Use multiplicity to predict crossing vs. bouncing
• Sketch complete polynomial graphs from algebra alone

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Read the zeros of a polynomial directly from its factored form
2. Determine multiplicity of each zero and explain crossing vs. bouncing behavior
3. Describe end behavior using the leading term (degree and leading coefficient)
4. Sketch a rough but structurally accurate polynomial graph
5. Interpret zeros in a real-world context

Recall: Zeros and the Zero-Product Property

• Zero-product property: if , then or
• Factor
• Factor Theorem: is a factor

Zeros of ?

From Zeros to the Graph

In APR.B.2, you found where a polynomial equals zero.

Now the question is: what does the graph do at those zeros?

• At a zero: the graph touches the -axis
• But does it cross through? Or bounce off?

The factored form tells you — but how?

Reading Zeros from Factored Form

If , then or (zero-product property)

Set each factor of equal to zero:

Y-intercept:

Worked Example: Zeros and Y-Intercept

Y-intercept:

Find the Sign Error Here

A student reads zeros from :

"Zeros: , , "

What is wrong? Give the correct zeros.

Hint: set the first factor equal to zero and solve.

Zeros in Context: What Do They Mean?

— height of a ball

Zeros: and

• : ball launched (height )
• : ball lands (height again)

Domain:

Check-In: Read Zeros and Y-Intercept

For :

1. What are the zeros of ?
2. What is the y-intercept?

Write your answers before moving on.

Multiplicity: How Many Times a Factor Appears

Exponent on factor = multiplicity. Odd → crosses. Even → bounces.

For :

Why Even Multiplicity Bounces: Sign Analysis

For , check the sign near :

Both negative → graph stays below -axis → bounces

Worked Example: Classify All Behaviors

— degree ,

Predict the Behavior, Then Verify Numerically

For :

Predict: will the graph cross or bounce at ?

Now verify: evaluate and .

Do both values have the same sign? What does that confirm?

Check-In: Zeros, Multiplicities, and Behavior

For :

1. What are the zeros and multiplicities?
2. At which zeros does the graph cross?
3. At which zeros does the graph bounce?

Also: what is the degree of ?

From Local to Global: End Behavior

Zeros → where. Multiplicity → how at each zero.

End behavior (the edges) depends only on the leading term:

End Behavior: The Four Possible Cases

Why the Leading Term Dominates Everything

For :

At : vs.

At : vs. — 1000× smaller

As grows, lower-degree terms become negligible.

Classify End Behavior: Three Examples

1. : degree 7, → ↓ ↑
2. : degree 4, → ↓ ↓
3. : degree , → ↑ ↑

For (3): degree = sum of exponents; = product of leading coefficients.

Check-In: State the End Behavior

State the end behavior for each:

For (3): find degree and leading coefficient first.

Five Steps for Sketching Any Polynomial

1. Zeros + multiplicities: set each factor
2. Y-intercept: evaluate
3. End behavior: degree + sign of
4. Plot zeros, y-intercept, arrows
5. Connect smoothly — cross (odd), bounce (even)

Full Sketch: Worked Example with All Features

Sketch

• Zeros: (crosses), (bounces), (crosses)
• Y-intercept:
• End behavior: degree , → ↓ ↓

Full Sketch: Degree Three, Fewer Features

Sketch

• Zeros: (crosses), (bounces)
• Y-intercept:
• End behavior: degree , → ↑ ↓

Upper left → cross at 0 → bounce at 3 → lower right

Rough Means Structurally Correct, Not Sloppy

Must be exact: zero locations, crossing vs. bouncing, end behavior

May be approximate: -values between intercepts, turning point heights

Structural errors = wrong sketch. Precision errors = rough sketch.

Which Sketch Is Structurally Correct?

For :

• Degree , leading coefficient → end behavior ↑ ↑
• Zeros: (crosses), (bounces), (crosses)

Two students drew different graphs. Which is correct?

Your Turn: Sketch This Polynomial Completely

Sketch completely.

• Degree: ___
• End behavior: ___
• Zeros and multiplicities: ___
• Y-intercept: ___
• Sketch

No hints. Use all five steps.

Three Errors That Ruin a Sketch

Factor ≠ zero: gives zero — a number, not an expression

Sign trap: gives zero , not

All zeros cross: bounces at — even multiplicity

Factored Form as a Graph X-Ray

✓ Zeros → -intercepts (set each factor )

✓ Multiplicity → crossing (odd) or bouncing (even) at each zero

✓ Leading term → end behavior (degree + sign of leading coefficient)

✓ Y-intercept → substitute

These four pieces produce a structurally accurate sketch with no calculator.

Next: These Skills Apply to Rational Functions

You can now sketch polynomial graphs using zeros, multiplicities, and end behavior.

The next domain: rational functions

• Zeros of → -intercepts (same as today)
• Zeros of → vertical asymptotes (new)
• End behavior analysis still applies

Same tools — new features.