Algebra 2 / Precalculus · Step-by-step guide

Polynomial Multiplicity

Understand repeated polynomial zeros and predict whether a graph crosses, touches, or flattens at each intercept.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A zero c has multiplicity m when (x−c)m\displaystyle (x-c)^m is a factor. Odd multiplicity crosses the x-axis; even multiplicity touches and turns. Larger multiplicities make the graph flatter near the zero.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Read factors as zeros

A factor x − c produces zero c; a factor x + c produces zero −c.

2

Parity controls crossing

Odd multiplicity changes the sign of the polynomial; even multiplicity keeps the same sign.

3

Multiplicities add to degree

The sum of multiplicities in a fully factored polynomial equals its degree.

02

See the structure

A picture makes the relationships easier to remember.

−22vertex (0, −1)
The coordinate graph marks the vertex and the two x-intercepts of a representative quadratic.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Even multiplicity

f(x)=(x−2)2(x+1)\displaystyle f(x)=(x-2)^2(x+1).

  1. x = 2 has multiplicity 2.
  2. Even multiplicity means the graph touches and turns.
  3. The graph crosses at x = −1.
AnswerTouch at 2; cross at −1
Example 2Odd multiplicity 3

g(x)=(x+4)3\displaystyle g(x)=(x+4)^3.

  1. The zero is −4.
  2. Multiplicity 3 is odd.
  3. The graph crosses with a flattened shape.
AnswerCrosses at −4
Example 3Find a missing multiplicity

A degree-7 polynomial has zeros of multiplicity 2 and 4 plus one more simple zero.

  1. Add known multiplicities: 2 + 4 = 6.
  2. Degree requires a total of 7.
  3. The remaining multiplicity is 1.
AnswerMultiplicity 1
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Reading x + 3 as the zero 3.

  • Saying all repeated zeros merely touch.

  • Ignoring multiplicity when sketching the graph.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Polynomial Graphs

Practice zeros, multiplicities, and graph shape.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

For f(x)=(x−2)3(x+1)2\displaystyle f(x)=(x-2)^3(x+1)^2, state each zero and multiplicity.

Show hint

Read each factor.

Show worked solution
  1. x−2=0\displaystyle x-2=0 gives 2 with exponent 3.
  2. x+1=0\displaystyle x+1=0 gives −1 with exponent 2.

Answer: 2 has multiplicity 3; −1 has multiplicity 2.

2

Does (x+4)2\displaystyle (x+4)^2 cross or touch at x = −4?

Show hint

Even multiplicities touch.

Show worked solution
  1. The exponent is 2, an even number.
  2. The sign does not change across the zero.

Answer: It touches and turns.

3

Does (x−1)5\displaystyle (x-1)^5 cross or touch at x = 1?

Show hint

Odd multiplicities cross.

Show worked solution
  1. The exponent 5 is odd.
  2. The graph changes sign and flattens near the zero.

Answer: It crosses and flattens.

4

Build a least-degree polynomial with zeros 3 (double) and −2 (single).

Show hint

Turn zeros into factors.

Show worked solution
  1. Zero 3 gives (x−3)2\displaystyle (x-3)^2.
  2. Zero −2 gives (x+2)\displaystyle (x+2).
  3. Multiply the factors or leave factored.

Answer: (x−3)2(x+2)\displaystyle (x-3)^2(x+2)

5

Find the degree of (x+1)4(x−5)3\displaystyle (x+1)^4(x-5)^3.

Show hint

Add factor exponents.

Show worked solution
  1. The factor degrees are 4 and 3.
  2. 4+3=7\displaystyle 4+3=7.

Answer: Degree 7.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A zero c has multiplicity m when (x−c)m\displaystyle (x-c)^m is a factor. Odd multiplicity crosses the x-axis; even multiplicity touches and turns. Larger multiplicities make the graph flatter near the zero.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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