What are the zeros of the given rational function below?

only
only
and
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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only
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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