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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 26
Chapter 4, Problem 26

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

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Identify the zeros of the polynomial function by setting each factor equal to zero. For the function \(f(x) = 3(x+5)(x+2)^2\), set \(x+5=0\) and \(x+2=0\) to find the zeros.
Solve each equation for \(x\): from \(x+5=0\), we get \(x=-5\); from \(x+2=0\), we get \(x=-2\). These are the zeros of the function.
Determine the multiplicity of each zero by looking at the exponent of the corresponding factor. The factor \((x+5)\) has an exponent of 1, so the zero \(x=-5\) has multiplicity 1. The factor \((x+2)^2\) has an exponent of 2, so the zero \(x=-2\) has multiplicity 2.
Interpret the multiplicity to understand the graph behavior at each zero: if the multiplicity is odd (like 1), the graph crosses the x-axis at that zero; if the multiplicity is even (like 2), the graph touches the x-axis and turns around at that zero.
Summarize the results: zero \(x=-5\) with multiplicity 1 means the graph crosses the x-axis at \(x=-5\), and zero \(x=-2\) with multiplicity 2 means the graph touches the x-axis and turns around at \(x=-2\).

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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 function 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 function equal to zero and solve for x, often by factoring or using other algebraic methods.
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Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a factor in the polynomial. For example, if (x + 2)^2 is a factor, then x = -2 is a zero with multiplicity 2. Multiplicity affects the behavior of the graph at that zero.
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Graph Behavior at Zeros Based on Multiplicity

The graph crosses the x-axis at zeros with odd multiplicity and touches the x-axis and turns around at zeros with even multiplicity. This means the shape of the graph near each zero depends on whether the multiplicity is odd or even.
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