Zeros and Multiplicity
Explore the concepts of zeros and their multiplicity in polynomial functions, including graphical interpretation and algebraic determination.
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Zeros and Multiplicity
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Read each question carefully and answer to the best of your ability. Show all your work for full credit.
1. What is the multiplicity of the zero x = 2 in the polynomial function f(x) = (x - 2)^3 (x + 1)^2?
1
2
3
5
2. If a polynomial function has a zero with an even multiplicity, how does the graph behave at that x-intercept?
It crosses the x-axis.
It touches the x-axis and turns around.
It is tangent to the x-axis.
Both b and c.
1. The of a zero is the number of times its corresponding factor appears in the factored form of the polynomial.
2. If a zero has an odd multiplicity, the graph of the polynomial will the x-axis at that point.
3. The sum of the multiplicities of all real zeros of a polynomial is less than or equal to the of the polynomial.
1. Find the zeros and their multiplicities for the polynomial function f(x) = x^4 - 2x^3 - 8x^2.
2. Describe the behavior of the graph of f(x) = (x + 3)(x - 1)^2 at each x-intercept.
1. A polynomial function can have more zeros than its degree.
True
False
2. If a polynomial function has a zero with multiplicity 1, the graph will have a local maximum or minimum at that zero.
True
False
Sketch a possible graph of a polynomial function with the following characteristics:
- Zeros at x = -2 (multiplicity 1), x = 0 (multiplicity 2), and x = 3 (multiplicity 3)