Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)4(x-3), the number 2 is a zero of multiplicity 4.

Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 3

Chapter 4, Problem 3

Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)⁴(x-3), the number 2 is a zero of multiplicity 4.

Verified step by step guidance

Recall that the zeros of a function are the values of \( x \) that make \( f(x) = 0 \).

Given \( f(x) = (x+2)^4 (x-3) \), the zeros come from setting each factor equal to zero: \( x+2=0 \) and \( x-3=0 \).

Solving these, we find the zeros are \( x = -2 \) and \( x = 3 \).

The multiplicity of a zero is the exponent on the factor corresponding to that zero. For \( x = -2 \), the multiplicity is 4 because of the \( (x+2)^4 \) term.

Since the problem states the number 2 is a zero of multiplicity 4, but the zero with multiplicity 4 is actually \( -2 \), the statement is false.

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Key Concepts

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

Zeros of a Function

A zero of a function is a value of x for which the function equals zero. To find zeros, set the function equal to zero and solve for x. Each zero corresponds to a root of the equation, indicating where the graph crosses or touches the x-axis.

Multiplicity of a Zero

Multiplicity refers to how many times a particular zero appears as a factor in the function. For example, if (x - a)^n is a factor, then x = a is a zero of multiplicity n. The multiplicity affects the graph's behavior at that zero, such as whether it crosses or just touches the x-axis.

Factoring and Identifying Zeros from Polynomial Expressions

To identify zeros and their multiplicities, express the polynomial in factored form. Each factor of the form (x - c)^k indicates a zero at x = c with multiplicity k. Understanding how to read and interpret these factors is essential for analyzing the function's roots.

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