What are the zeros of the given rational function below? \( f(x)=\frac{x^{2}}{x^{3}-6 x^{2}+5 x} \) \( x=0 \) No zeros possible for this function \( x=-6 \) \( x=5 \)

To determine the zeros of the rational function \( f(x) = \frac{x^{2}}{x^{3} - 6x^{2} + 5x} \), we need to find the values of \( x \) that make the numerator zero **and** ensure that the denominator is not zero at those points. 1. **Find when the numerator is zero:** \[ x^{2} = 0 \implies x = 0 \] 2. **Check the denominator at \( x = 0 \):** \[ x^{3} - 6x^{2} + 5x = 0^{3} - 6(0)^{2} + 5(0) = 0 \] Since both the numerator and denominator are zero at \( x = 0 \), this point is not a valid zero of the function—it indicates a potential hole or vertical asymptote, not a zero. 3. **Conclusion:** Since the only candidate \( x = 0 \) does not satisfy the condition (denominator is also zero), the function \( f(x) \) **has no zeros**. **Answer:** No zeros possible for this function