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Identify Zeros and Multiplicities of Zeros: f(x) = -2x4(x+1)°(x-2)²

Identify Zeros and Multiplicities of Zeros: f(x) = -2x4(x+1)°(x-2)²

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Identifying Zeros and Multiplicities of Zeros** Consider the function \( f(x) = -2x^4(x+1)^3(x-2)^2 \). This function is a polynomial, and we aim to identify its zeros and their respective multiplicities. 1. **Zero at \( x = 0 \):** - The term \( x^4 \) contributes a zero at \( x = 0 \). - The multiplicity of this zero is 4, due to the exponent of the term. 2. **Zero at \( x = -1 \):** - The term \( (x+1)^3 \) contributes a zero at \( x = -1 \). - The multiplicity of this zero is 3, as indicated by the exponent. 3. **Zero at \( x = 2 \):** - The term \( (x-2)^2 \) contributes a zero at \( x = 2 \). - The multiplicity of this zero is 2, shown by the exponent. **Understanding Multiplicities:** - The *multiplicity* of a zero refers to the number of times the zero appears in the factored form of the polynomial. - A zero with even multiplicity indicates the graph touches the x-axis but does not cross it. - A zero with odd multiplicity means the graph crosses the x-axis at that point.
Transcribed Image Text:**Identifying Zeros and Multiplicities of Zeros** Consider the function \( f(x) = -2x^4(x+1)^3(x-2)^2 \). This function is a polynomial, and we aim to identify its zeros and their respective multiplicities. 1. **Zero at \( x = 0 \):** - The term \( x^4 \) contributes a zero at \( x = 0 \). - The multiplicity of this zero is 4, due to the exponent of the term. 2. **Zero at \( x = -1 \):** - The term \( (x+1)^3 \) contributes a zero at \( x = -1 \). - The multiplicity of this zero is 3, as indicated by the exponent. 3. **Zero at \( x = 2 \):** - The term \( (x-2)^2 \) contributes a zero at \( x = 2 \). - The multiplicity of this zero is 2, shown by the exponent. **Understanding Multiplicities:** - The *multiplicity* of a zero refers to the number of times the zero appears in the factored form of the polynomial. - A zero with even multiplicity indicates the graph touches the x-axis but does not cross it. - A zero with odd multiplicity means the graph crosses the x-axis at that point.
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