Question
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 \( x=-6 \) \( x=5 \)
Ask by Hardy Vargas. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
There are no zeros for this function.
Solution
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
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To find the zeros of the rational function \( f(x) = \frac{x^2}{x^3 - 6x^2 + 5x} \), we need to set the numerator equal to zero. The numerator \( x^2 \) equals zero when \( x = 0 \). Therefore, \( x = 0 \) is the only zero of the function. The denominator, \( x^3 - 6x^2 + 5x \), is important for the function's behavior but does not affect the zeros. One common mistake in identifying zeros is overlooking the influence of the denominator. The function is undefined wherever the denominator equals zero, so while we focus on the zeros from the numerator, it's equally crucial to recognize that those points where the denominator is zero indicate vertical asymptotes, not zeros of the function.
