Zeros and Multiplicity Worksheet
Relate repeated factors to zeros and local graph behavior. Includes a worked example, printable practice, and an answer key with explanations.
The answer key follows the practice below. In the editor, remove the final answer-key section before sharing a student copy.
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Zeros and Multiplicity
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Learning goal and worked example
Relate repeated factors to zeros and local graph behavior.
For (x−1)²(x+4), x=1 has multiplicity 2 and the graph touches and turns; x=−4 has multiplicity 1 and the graph crosses.
For a nonzero polynomial, odd multiplicity crosses and even multiplicity turns. Multiplicity is the exponent of the corresponding factor.
Read each question carefully and provide your answers in the space provided. Show all your work for full credit.
1. What is the multiplicity of the zero x = 3 in the polynomial function f(x) = (x-3)^2 (x+1)?
1
2
3
4
2. If a zero has an even multiplicity, how does the graph of the polynomial function behave at the x-axis?
It crosses the x-axis
It touches the x-axis and turns around
It has a vertical asymptote
It has a hole
3. The of a zero is the number of times its corresponding factor appears in the factored form of the polynomial.
4. If a zero has an odd multiplicity, the graph of the polynomial function the x-axis at that point.
5. Find the zeros and their multiplicities for the polynomial function f(x) = x^3 - 4x^2 + 4x.
6. Describe the behavior of the graph of f(x) = (x+2)^3 (x-1)^2 at each of its zeros.
7. Sketch a qualitative graph with zeros at -3 (multiplicity 1), 0 (multiplicity 2), and 2 (multiplicity 3). Choose a positive leading coefficient and label both ends. Use your own vertical scale; exact turning-point heights are not required.
Answer key and worked solutions
1. b (multiplicity 2). 2. b (touches and turns around).
3. Multiplicity. 4. Crosses.
5. x³−4x²+4x=x(x−2)²: zero 0 has multiplicity 1; zero 2 has multiplicity 2.
6. At −2, multiplicity 3: crosses with a flattened tangent. At 1, multiplicity 2: touches and turns.
7. One model is f(x)=(x+3)x²(x−2)³. Degree 6 with positive leading coefficient gives both ends upward. Cross at −3, touch and turn at 0, and cross with flattening at 2. Signs from left to right are positive, negative, negative, positive across those zeros. Accept positive constant multiples and qualitative sketches meeting these conditions.
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