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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 16
Chapter 4, Problem 16

Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−2(x−1)(x+2)2(x+5)3f(x)=-2(x-1)(x+2)^2(x+5)^3

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1
Identify the zeros of the polynomial by setting each factor equal to zero: solve \(x - 1 = 0\), \(x + 2 = 0\), and \(x + 5 = 0\) to find the zeros.
Determine the multiplicity of each zero by looking at the exponent on each factor: the factor \((x - 1)\) has multiplicity 1, \((x + 2)^2\) has multiplicity 2, and \((x + 5)^2\) has multiplicity 2.
Recall that if a zero has an odd multiplicity, the graph crosses the x-axis at that zero; if the multiplicity is even, the graph touches the x-axis and turns around at that zero.
Summarize the behavior at each zero: for \(x = 1\) (multiplicity 1), the graph crosses the x-axis; for \(x = -2\) and \(x = -5\) (both multiplicity 2), the graph touches and turns around at the x-axis.
Combine all this information to describe the zeros, their multiplicities, and the behavior of the graph at each zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Zeros of a Polynomial Function

Zeros of a polynomial are the values of x that make the function equal to zero. They correspond to the x-intercepts of the graph. To find zeros, set the polynomial equal to zero and solve for x, often by factoring or using given factored form.
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Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears as a factor in the polynomial. If a zero has even multiplicity, the graph touches the x-axis and turns around at that zero. If the multiplicity is odd, the graph crosses the x-axis at that zero.
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Behavior of Graph at Zeros

The graph's behavior at each zero depends on the zero's multiplicity. For odd multiplicity, the graph crosses the x-axis, indicating a sign change. For even multiplicity, the graph only touches the axis and reverses direction, showing no sign change.
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Related Practice
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