What are the zeros of the given rational function below?

and
only
only
No zeros possible for this function

Ask by Osborne Gibson. in the United States

Jan 25,2025

Question

What are the zeros of the given rational function below? $$ f(x)=\frac{-x^{2}+16 x-63}{x^{2}-2 x-35} $$ $$ x=9 $$ and $$ x=5 $$ $$ x=9 $$ only $$ x=5 $$ only No zeros possible for this function

UpStudy AI · Accepted Answer

To determine the zeros of the rational function $$ f(x) = \frac{-x^{2} + 16x - 63}{x^{2} - 2x - 35} $$, follow these steps:

1. **Find the Zeros of the Numerator:**
   Zeros occur where the numerator is equal to zero.
   $$
   -x^{2} + 16x - 63 = 0
   $$
   Multiply both sides by $$-1$$ to simplify:
   $$
   x^{2} - 16x + 63 = 0
   $$
   Factor the quadratic:
   $$
   (x - 9)(x - 7) = 0
   $$
   Therefore, the potential zeros are $$ x = 9 $$ and $$ x = 7 $$.

2. **Check the Denominator for Restrictions:**
   The denominator cannot be zero, as this would make the function undefined.
   $$
   x^{2} - 2x - 35 = 0
   $$
   Factor the quadratic:
   $$
   (x - 7)(x + 5) = 0
   $$
   So, $$ x = 7 $$ and $$ x = -5 $$ are not in the domain of $$ f(x) $$.

3. **Determine the Valid Zeros:**
   - $$ x = 9 $$ is a valid zero since it does not make the denominator zero.
   - $$ x = 7 $$ is **not** a valid zero because it makes the denominator zero, resulting in an undefined value.

**Conclusion:**
The only zero of the function is $$ x = 9 $$.

**Answer:**  
$$ x=9 $$ only
The only zero of the function is $$ x = 9 $$.

What are the zeros of the given rational function below?

and
only
only
No zeros possible for this function

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