Study Notes
Algebraic methods involve expressing rational functions in partial fractions and decomposing them, especially when dealing with improper fractions. Improper algebraic fractions can be written as the sum of a polynomial and a proper algebraic fraction.
- Algebraic Fractions — Functions of the form f(x)/g(x), where f(x) and g(x) are polynomials. Example: (x^2 + 3x + 2)/(x + 1)
- Improper Fractions — Fractions where the degree of the numerator is greater than or equal to the degree of the denominator. Example: (x^3 + 2x^2 + 3)/(x^2 + 1)
- Partial Fractions — Splitting a single fraction into separate fractions with linear or quadratic factors as denominators. Example: (x^2 + 3x + 2)/(x + 1)(x + 2) = A/(x + 1) + B/(x + 2)
Exam Tips
Key Definitions to Remember
- Algebraic Fractions: Functions of the form f(x)/g(x)
- Improper Fractions: Degree of numerator ≥ degree of denominator
- Partial Fractions: Splitting a fraction into simpler fractions
Common Confusions
- Forgetting to express improper fractions as the sum of a polynomial and a proper fraction
- Mixing up the rules for splitting fractions with linear and quadratic factors
Typical Exam Questions
- How do you express (x^3 + 2x^2 + 3)/(x^2 + 1) in partial fractions? First, divide to express as a polynomial plus a proper fraction, then split the proper fraction.
- What is the partial fraction decomposition of (x^2 + 3x + 2)/(x + 1)(x + 2)? A/(x + 1) + B/(x + 2)
- How do you handle repeated linear factors in partial fractions? Use separate terms for each power of the repeated factor.
What Examiners Usually Test
- Ability to decompose improper fractions into partial fractions
- Correct application of rules for partial fraction decomposition
- Understanding of when to use polynomial division in improper fractions