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