Question
What are the zeros of the given rational function below? \( f(x)=\frac{x^{3}-4 x^{2}}{x^{2}-9} \) \( x=0 \) only \( x=4 \) only \( x=0 \) and \( x=4 \) No zeros possible for this function
Ask by Fernandez Reid. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The zeros of the function are \( x = 0 \) and \( x = 4 \).
Solution
To determine the zeros of the rational function \( f(x) = \frac{x^{3} - 4x^{2}}{x^{2} - 9} \), we need to find the values of \( x \) that make the numerator zero while ensuring the denominator is not zero.
1. **Find the zeros of the numerator:**
\[
x^{3} - 4x^{2} = 0 \\
x^{2}(x - 4) = 0
\]
This gives:
\[
x = 0 \quad \text{and} \quad x = 4
\]
2. **Check the denominator for these values:**
\[
x^{2} - 9 = (x + 3)(x - 3)
\]
- For \( x = 0 \):
\[
0^{2} - 9 = -9 \neq 0
\]
- For \( x = 4 \):
\[
4^{2} - 9 = 16 - 9 = 7 \neq 0
\]
Both \( x = 0 \) and \( x = 4 \) make the numerator zero without making the denominator zero. Therefore, these are the zeros of the function.
**Answer:** \( x = 0 \) and \( x = 4 \)
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Mind Expander
To find the zeros of the function \( f(x)=\frac{x^{3}-4 x^{2}}{x^{2}-9} \), we need to focus on the numerator since the zeros of a rational function occur when the numerator equals zero (while ensuring the denominator isn't zero at those points). The numerator, \( x^3 - 4x^2 \), can be factored as \( x^2(x - 4) \), revealing that the zeros are \( x = 0 \) and \( x = 4 \). However, we must check that these values do not make the denominator zero. The denominator, \( x^2 - 9 \), is zero at \( x = 3 \) and \( x = -3 \), which are not our zeros, so both \( x = 0 \) and \( x = 4 \) are valid zeros of the function. In summary, the zeros of the function are \( x=0 \) and \( x=4 \).
