f(x)=\frac{x^{2}-25}{2x^{2}+14x+20} (find hole)

To find the hole of the function \( f(x) = \frac{x^2 - 25}{2x^2 + 14x + 20} \), we follow these steps:

### Step 1: Factor both the numerator and the denominator
- **Numerator**: \( x^2 - 25 \) is a difference of squares and can be factored as:
  \[
  x^2 - 25 = (x - 5)(x + 5)
  \]
  
- **Denominator**: \( 2x^2 + 14x + 20 \) can be factored by first factoring out a 2:
  \[
  2x^2 + 14x + 20 = 2(x^2 + 7x + 10)
  \]
  Now, factor \( x^2 + 7x + 10 \):
  \[
  x^2 + 7x + 10 = (x + 5)(x + 2)
  \]
  So, the denominator becomes:
  \[
  2(x + 5)(x + 2)
  \]

### Step 2: Rewrite the function
Now rewrite \( f(x) \) with the factored forms:
\[
f(x) = \frac{(x - 5)(x + 5)}{2(x + 5)(x + 2)}
\]

### Step 3: Simplify the function
Cancel out the common factor \( (x + 5) \) from both the numerator and the denominator:
\[
f(x) = \frac{x - 5}{2(x + 2)}
\]

### Step 4: Identify the hole
The factor \( (x + 5) \) was canceled, but it originally caused the denominator to be 0 when \( x = -5 \). Therefore, there is a **hole** at \( x = -5 \).

To find the \( y \)-coordinate of the hole, substitute \( x = -5 \) into the simplified function \( f(x) = \frac{x - 5}{2(x + 2)} \):
\[
f(-5) = \frac{-5 - 5}{2(-5 + 2)} = \frac{-10}{2(-3)} = \frac{-10}{-6} = \frac{5}{3}
\]

### Conclusion:
There is a hole at \( x = -5 \), and its coordinates are \( (-5, \frac{5}{3}) \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find vertical asymptotes of rational functions?
2. What is the difference between holes and vertical asymptotes?
3. How can we determine the domain of a rational function?
4. How do you find horizontal asymptotes of rational functions?
5. What role do canceled factors play in determining the hole of a function?

**Tip:** When simplifying rational functions, always check for common factors in both the numerator and denominator to identify potential holes.

Step-by-step guide to finding the hole of the rational function f(x) = (x^2 - 25) / (2x^2 + 14x + 20). Learn how to factorize the numerator and denominator, simplify the function, and identify the hole. Suitable for high school students in Grades 10-12.

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