Give the domain of the function.
f left parenthesis x right parenthesis equals StartFraction x Superscript 4 Baseline plus 9 Over x squared plus 3 x minus 40 EndFraction

The given function is:

\[
f(x) = \frac{x^4 + 9}{x^2 + 3x - 40}
\]

To determine the domain of the function, we need to find the values of \( x \) for which the function is defined. Specifically, the function will be undefined where the denominator equals zero, because division by zero is undefined.

### Step 1: Set the denominator equal to zero
The denominator is:

\[
x^2 + 3x - 40
\]

We need to solve the equation:

\[
x^2 + 3x - 40 = 0
\]

### Step 2: Solve the quadratic equation
We can solve this quadratic equation using factoring. We are looking for two numbers that multiply to \(-40\) and add up to \(3\). These numbers are \(8\) and \(-5\), so we can factor the quadratic as follows:

\[
x^2 + 3x - 40 = (x + 8)(x - 5)
\]

### Step 3: Find the values of \( x \)
Now, set each factor equal to zero and solve for \( x \):

\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

### Step 4: State the domain
The function is undefined at \( x = -8 \) and \( x = 5 \) because these values make the denominator zero. Therefore, the domain of the function is all real numbers except \( x = -8 \) and \( x = 5 \).

In interval notation, the domain is:

\[
(-\infty, -8) \cup (-8, 5) \cup (5, \infty)
\]

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Do you need further clarification or details?

Here are 5 related questions to deepen your understanding:
1. How do you find the domain of a function with a square root in the denominator?
2. What is the general strategy to find the domain of rational functions?
3. How does factoring help in solving quadratic equations?
4. What would happen if the numerator also had a zero at \(x = -8\) or \(x = 5\)?
5. How do vertical asymptotes relate to the domain of a rational function?

**Tip:** Always check the denominator of a rational function to ensure it does not equal zero, as this determines the points where the function is undefined.

Give the domain of the function. f(x) = (x^4 + 9) / (x^2 + 3x - 40)

Explore how to find the domain of the rational function (x^4 + 9) / (x^2 + 3x - 40) by identifying values that make the denominator zero and applying algebraic factoring techniques. Perfect for high school students (Grades 9-12) learning about rational functions and quadratic equations.

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