(a) Find the rational zeros and then the other zeros of the polynomial function $f(x)=3 x^{3}+5 x^{2}+75 x+125$, that is, solve $f(x)=0$.
(b) Factor $f(x)$ into linear factors.
(a) Select the correct choice below and fill in any answer box(es) within your choice.
(Type an exact answer, using radicals and $i$ as needed. Use integers or fractions for any numbers in the expression. Use a comma to

Step 1 ：First, we need to find the rational roots of the polynomial function $f(x)=3 x^{3}+5 x^{2}+75 x+125$. According to the rational root theorem, if a polynomial has a rational root, then it must be a factor of the constant term divided by a factor of the leading coefficient. In this case, the constant term is 125 and the leading coefficient is 3.

Step 2 ：We find the factors of 125 and 3, and then divide each factor of 125 by each factor of 3 to get the possible rational roots.

Step 3 ：After finding the rational roots, we can use synthetic division or polynomial division to divide the polynomial by the factors corresponding to the rational roots to get a quadratic polynomial.

Step 4 ：Then, we can use the quadratic formula to find the other roots of the polynomial.

Step 5 ：The rational roots of the polynomial are $-\frac{5}{3}$. The other roots are $-5i$ and $5i$, which are complex roots. Now, we have found all the roots of the polynomial.

Step 6 ：Final Answer: The rational zeros of the polynomial function $f(x)=3 x^{3}+5 x^{2}+75 x+125$ are \(\boxed{-\frac{5}{3}}\). The other zeros are \(\boxed{-5i}\) and \(\boxed{5i}\).