Step 1 :We are given the polynomial function \(f(x)=(x-5)^{3}(x^{2}-2)\). We need to find all zeros of this function and their multiplicities.
Step 2 :To find the zeros of the polynomial, we set the polynomial equal to zero and solve for x. These solutions can be real or complex numbers.
Step 3 :Setting each factor equal to zero, we find that the roots of the polynomial are \(5\), \(\sqrt{2}\), and \(-\sqrt{2}\).
Step 4 :We also observe that the root \(5\) is repeated three times, indicating that its multiplicity is 3.
Step 5 :Final Answer: The zeros of the polynomial are \(\boxed{5, \sqrt{2}, -\sqrt{2}}\). The zero \(5\) has a multiplicity of 3.