1. **State the problem:** Find the polynomial function with real coefficients that has zeros at $0$, $3$, and $3+i$. 2. **Recall the rule for complex roots:** If a polynomial has real coefficients and a complex root $a+bi$, then its conjugate $a-bi$ is also a root. Since $3+i$ is a root, $3 - i$ must also be a root. 3. **List all roots:** $0$, $3$, $3+i$, and $3 - i$. 4. **Write the polynomial as a product of factors:** $$f(x) = x(x - 3)(x - (3+i))(x - (3 - i))$$ 5. **Multiply the complex conjugate factors:** $$ (x - (3+i))(x - (3 - i)) = ((x - 3) - i)((x - 3) + i) = (x - 3)^2 - i^2 = (x - 3)^2 + 1 $$ 6. **Expand $(x - 3)^2 + 1$:** $$ (x - 3)^2 + 1 = (x^2 - 6x + 9) + 1 = x^2 - 6x + 10 $$ 7. **Write the full polynomial:** $$ f(x) = x(x - 3)(x^2 - 6x + 10) $$ 8. **Expand step-by-step:** First multiply $x$ and $(x - 3)$: $$ x(x - 3) = x^2 - 3x $$ Then multiply by $(x^2 - 6x + 10)$: $$ (x^2 - 3x)(x^2 - 6x + 10) = x^2(x^2 - 6x + 10) - 3x(x^2 - 6x + 10) $$ $$ = x^4 - 6x^3 + 10x^2 - 3x^3 + 18x^2 - 30x $$ 9. **Combine like terms:** $$ x^4 - 6x^3 - 3x^3 + 10x^2 + 18x^2 - 30x = x^4 - 9x^3 + 28x^2 - 30x $$ 10. **Final polynomial function:** $$ \boxed{f(x) = x^4 - 9x^3 + 28x^2 - 30x} $$ This polynomial has real coefficients and the given zeros.