📚 Topic Summary
Partial fraction decomposition is a technique used to break down a rational function into simpler fractions. When dealing with distinct linear factors in the denominator, each factor corresponds to a simple fraction. For example, if you have a fraction like $\frac{P(x)}{(x-a)(x-b)}$, it can be decomposed into $\frac{A}{x-a} + \frac{B}{x-b}$, where A and B are constants that need to be determined.
This method simplifies integration and other algebraic manipulations by allowing us to work with easier-to-manage individual fractions. The key is to correctly identify the distinct linear factors and solve for the unknown constants in the numerators of the decomposed fractions.
🧠 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Partial Fraction |
A. The process of breaking down a rational function into simpler fractions. |
| 2. Decomposition |
B. A fraction that is part of the sum representing the original rational function. |
| 3. Linear Factor |
C. An algebraic expression of the form $ax + b$. |
| 4. Rational Function |
D. A function that can be expressed as the quotient of two polynomials. |
| 5. Constant |
E. A fixed value that does not change. |
✏️ Part B: Fill in the Blanks
Partial fraction decomposition is used to simplify ______ functions. When the denominator has distinct ______ factors, we can express the original fraction as a sum of simpler fractions, each with one of these factors in its ______. The goal is to find the unknown ______ in the numerators.
🤔 Part C: Critical Thinking
Explain in your own words why partial fraction decomposition is a useful technique in calculus, especially when integrating rational functions. Give a specific example of a rational function with distinct linear factors and describe how you would decompose it.