Problem

Given the function f(x) = (x - 2)^2 (x + 3)^3, identify the zeros of the function and their multiplicities.

Answer

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Answer

The multiplicity of a zero is the power of the factor in the factored form of the function. So the zero 2 has multiplicity 2 and the zero -3 has multiplicity 3.

Steps

Step 1 ：The zeros of a function are the x-values that make the function equal to zero. So we set the function equal to zero: (x−2)2(x+3)3=0

Step 2 ：A product is zero if and only if one (or more) of the factors is zero. So we can set each factor equal to zero: (x−2)2=0 and (x+3)3=0

Step 3 ：Solving these equations gives the zeros: x=2 and x=−3

Step 4 ：The multiplicity of a zero is the power of the factor in the factored form of the function. So the zero 2 has multiplicity 2 and the zero -3 has multiplicity 3.

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