Zeros and Multiplicity
This worksheet focuses on understanding zeros of polynomial functions and their multiplicities, including how they affect graph behavior.
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Zeros and Multiplicity
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Read each question carefully and provide thorough answers. Show all your work for full credit.
1. A zero of a polynomial function f(x) is a value 'c' such that f(c) = .
2. The multiplicity of a zero is the number of times its corresponding appears in the factored form of the polynomial.
3. If a zero has an even multiplicity, the graph of the function will the x-axis at that zero.
4. If a zero has an odd multiplicity, the graph of the function will through the x-axis at that zero.
5. For the polynomial function f(x) = (x - 2)^3 (x + 1)^2 (x - 5):
a. List all the zeros:
b. State the multiplicity of each zero:
6. Write a polynomial function in factored form with the following characteristics:
- A zero at x = 3 with multiplicity 2
- A zero at x = -4 with multiplicity 1
- A zero at x = 0 with multiplicity 3
7. If the graph of a polynomial function touches the x-axis at x = -2 and turns around, what can be inferred about the multiplicity of the zero at x = -2?
It has an odd multiplicity.
It has an even multiplicity.
It has a multiplicity of 1.
The multiplicity cannot be determined.
8. Describe the end behavior and the behavior at each zero for the function f(x) = -(x + 3)(x - 1)^2.
9. A polynomial function can have more zeros than its degree.
True
False
10. The multiplicity of a zero affects the number of x-intercepts of the graph.
True
False
11. If a polynomial function has zeros at x = -1 (multiplicity 3), x = 2 (multiplicity 1), and x = 4 (multiplicity 2), describe what happens to the graph at each of these zeros.