What are the zeros of the given rational function below?

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only
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No zeros possible for this function

Ask by Fernandez Reid. in the United States

Jan 25,2025

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

UpStudy AI · Accepted Answer

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 $$ x = 4 $$
The zeros of the function are $$ x = 0 $$ and $$ x = 4 $$.

What are the zeros of the given rational function below?

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No zeros possible for this function

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What are the zeros of the given rational function below? only only and No zeros possible for this function