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Chapter 2: Problem 1

Find the zeros of the polynomial function and state the multiplicity of each zero. $$P(x)=(x-3)^{2}(x+5)$$

Short Answer

Expert verified
The zeros of the polynomial function \(P(x)=(x-3)^{2}(x+5)\) are \(x=3\) with multiplicity 2, and \(x=-5\) with multiplicity 1.

Step by step solution

01

Find the Zeros

To find the zeros of the polynomial, it is necessary to solve the equation \(P(x)=0\), i.e., \((x-3)^{2}(x+5)=0\). The equation will be equal to zero when either \((x-3)^{2}=0\) or \(x+5=0\). Solve both of these for \(x\).
02

Find the Zero from first part

By solving \((x-3)^{2}=0\), it is clear that \(x=3\) is a zero, because \((3-3)^{2}=0\).
03

Find the Zero from second part

By solving \(x+5=0\), it is clear that \(x=-5\) is a zero, because \((-5+5)=0\).
04

Identify the Multiplicity of each Zero

The multiplicity of the zeros is given by the exponents of the respective terms. Thus, the multiplicity of \(x=3\) is 2 (as it's given by \((x-3)^2\)) and the multiplicity of \(x=-5\) is 1 (as it's given by \(x+5\)), as there is no explicit exponent associated with it.

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Most popular questions from this chapter

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{4}+x^{3}-3 x^{2}-5 x-2$$

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