Given the function \(f(x) = (x + 1)(x - 2)^2(x + 3)^3\), identify the zeros and their multiplicities.
Answer
The multiplicity of a root is the number of times it appears in the factored form. In this case, \(-1\) and \(2\) and \(-3\) are the roots, and their multiplicities are 1, 2, and 3 respectively, as indicated by the exponents of the corresponding factors.
Step 1 ：Firstly, set \(f(x)\) to zero and solve for \(x\): \[0 = (x + 1)(x - 2)^2(x + 3)^3\]
Step 2 ：The roots of this equation are the values of \(x\) that make the equation true. Here, the roots are \(-1\), \(2\), and \(-3\), which are obtained by setting each factor equal to zero.
Step 3 ：The multiplicity of a root is the number of times it appears in the factored form. In this case, \(-1\) and \(2\) and \(-3\) are the roots, and their multiplicities are 1, 2, and 3 respectively, as indicated by the exponents of the corresponding factors.
