write out the form of the partial fraction decomposition of the function. x-30/x^2+x-30 and the function 1/x^2+x^4

Let's decompose the given rational functions into partial fractions. I will show the form of the decomposition for both.

### 1. Function: \(\frac{x - 30}{x^2 + x - 30}\)

First, we factor the denominator:

\[
x^2 + x - 30 = (x - 5)(x + 6)
\]

Thus, the function becomes:

\[
\frac{x - 30}{(x - 5)(x + 6)}
\]

Now, the partial fraction decomposition will be in the form:

\[
\frac{x - 30}{(x - 5)(x + 6)} = \frac{A}{x - 5} + \frac{B}{x + 6}
\]

Where \(A\) and \(B\) are constants to be determined.

### 2. Function: \(\frac{1}{x^2 + x^4}\)

First, factor the denominator:

\[
x^2 + x^4 = x^2(1 + x^2)
\]

So, the function becomes:

\[
\frac{1}{x^2(1 + x^2)}
\]

The partial fraction decomposition will take the form:

\[
\frac{1}{x^2(1 + x^2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{1 + x^2}
\]

Where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

---

Would you like details on solving for these constants?

Here are some related questions you can explore:
1. How do you solve for the constants \(A\) and \(B\) in partial fraction decomposition?
2. What is the method to factor a quadratic equation like \(x^2 + x - 30\)?
3. How does the presence of irreducible quadratic terms like \(1 + x^2\) affect the decomposition?
4. What is the difference between decomposing a rational function with distinct linear factors and repeated factors?
5. How does partial fraction decomposition help in integrating rational functions?

**Tip**: Always start by factoring the denominator as much as possible when doing partial fraction decomposition.

Learn how to perform partial fraction decomposition on the functions (x - 30) / (x^2 + x - 30) and 1 / (x^2 + x^4). This detailed solution covers factoring quadratic equations and decomposing rational functions, suitable for advanced algebra or pre-calculus students.

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