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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

Why it matters: Polynomial models describe smooth turning behavior in design, approximation, data fitting, and STEM modeling.

Worked example

Problem. In p(x) = (x - 3)^2(x - 7)^3(x - 10)^4, what is the multiplicity of the zero x = 3?

  1. Multiplicity is the exponent on the matching factor.
  2. The factor (x - 3) is raised to the power 2.
  3. So the zero x = 3 has multiplicity 2.

Answer: 2

Practice problems

1. In p(x) = (x - 3)^2(x - 7)^3(x - 10)^4, what is the multiplicity of the zero x = 3?

Show solution
  1. Multiplicity is the exponent on the matching factor.
  2. The factor (x - 3) is raised to the power 2.
  3. So the zero x = 3 has multiplicity 2.

Answer: 2

2. In p(x) = (x - 4)^1(x - 8)^4(x - 11)^3, the factor (x - 8) is raised to what power?

Show solution
  1. The exponent on a factor is the multiplicity of its zero.
  2. Find (x - 8) and read its exponent: it is raised to the power 4.
  3. So the factor (x - 8) is raised to the power 4.

Answer: 4

3. For p(x) = (x - 5)^2(x - 6)^2(x - 12)^4, the degree equals the sum of all multiplicities. Find the degree.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For p(x) = (x - 5)^2(x - 6)^2(x - 12)^4, the degree equals the sum of all multiplicities. Find the degree.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Add the exponents: 2 + 2 + 4.
  4. 2 + 2 + 4 = 8.
  5. The sum of multiplicities is the degree, so the degree is 8.
  6. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

4. How many DISTINCT zeros does p(x) = (x - 2)^1(x - 7)^3(x - 13)^3 have?

Show solution
  1. Count the different factors, not the exponents.
  2. The distinct factors are (x - 2), (x - 7), and (x - 13).
  3. So there are 3 distinct zeros.

Answer: 3

5. How many DISTINCT zeros does p(x) = (x - 3)^2(x - 8)^4 have?

Show solution
  1. Count the different factors, ignoring exponents.
  2. The distinct factors are (x - 3) and (x - 8).
  3. So there are 2 distinct zeros.

Answer: 2

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