Zeros and Multiplicity
A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.
Odd multiplicity usually crosses the x-axis; even multiplicity usually touches and turns. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Use zeros and multiplicity to predict crossing or touching behavior
- Use zeros and multiplicity in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. A zero with odd multiplicity usually makes the graph:
- Odd multiplicity changes sign across the zero.
- The graph crosses the x-axis.
- Higher odd multiplicity may flatten as it crosses.
Answer: cross the x-axis
Practice problems
1. A zero with odd multiplicity usually makes the graph:
Choices: cross the x-axis · touch and turn · make a hole · become undefined
Show solution
- Odd multiplicity changes sign across the zero.
- The graph crosses the x-axis.
- Higher odd multiplicity may flatten as it crosses.
Answer: cross the x-axis
2. A zero with odd multiplicity usually makes the graph: (variation 2)
Choices: cross the x-axis · touch and turn · make a hole · become undefined
Show solution
- Odd multiplicity changes sign across the zero.
- The graph crosses the x-axis.
- Higher odd multiplicity may flatten as it crosses.
Answer: cross the x-axis
3. A zero with even multiplicity usually makes the graph:
Choices: touch and turn at the x-axis · cross sharply · create a vertical asymptote · remove the y-intercept
Show solution
- Core Practice: First identify exactly what the question is asking: A zero with even multiplicity usually makes the graph:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Even multiplicity keeps the sign the same.
- The graph touches and turns around.
- It does not cross at that zero.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: touch and turn at the x-axis
4. A zero with even multiplicity usually makes the graph: (variation 2)
Choices: touch and turn at the x-axis · cross sharply · create a vertical asymptote · remove the y-intercept
Show solution
- Core Practice: First identify exactly what the question is asking: A zero with even multiplicity usually makes the graph:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Even multiplicity keeps the sign the same.
- The graph touches and turns around.
- It does not cross at that zero.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: touch and turn at the x-axis
5. In p(x) = (x - 2)^3(x + 1)^2, the zero x = 2 has multiplicity:
Choices: 3 · 2 · 1 · 5
Show solution
- Core Practice: First identify exactly what the question is asking: In p(x) = (x - 2)^3(x + 1)^2, the zero x = 2 has multiplicity:
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Multiplicity is the exponent on the factor.
- The factor x - 2 has exponent 3.
- So the multiplicity is 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 3
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