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 zeros directly from factored form
2. Find multiplicity and explain crossing vs. bouncing
3. Describe end behavior from the leading term
4. Sketch a rough, structurally accurate graph
5. Interpret zeros in 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. — a million times smaller

As grows, the constant term becomes 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.