Partial fractions

Summary

Partial fractions involve expressing a single fraction with multiple linear factors in the denominator as separate fractions with those factors as denominators. If the fraction is improper, it must first be expressed as the sum of a polynomial and a proper fraction before decomposition.

• Partial Fractions — A method to express a fraction as a sum of simpler fractions. Example: $\frac{1}{(x+1)(x+2)}$ can be split into $\frac{A}{x+1} + \frac{B}{x+2}$.
• Improper Fraction — A fraction where the degree of the numerator is greater than or equal to the degree of the denominator. Example: $\frac{x^2 + 3x + 2}{x+1}$ is improper and needs simplification.
• Repeated Factors — When a factor in the denominator appears more than once. Example: $\frac{1}{(x+1)^2}$ requires special handling in partial fractions.

Exam Tips

Key Definitions to Remember

• Partial fractions involve breaking down a complex fraction into simpler parts.
• An improper fraction has a higher degree in the numerator than the denominator.

Common Confusions

• Forgetting to simplify an improper fraction before decomposition.
• Misidentifying repeated factors in the denominator.

Typical Exam Questions

• How do you express $\frac{1}{(x+1)(x+2)}$ in partial fractions? Use $\frac{A}{x+1} + \frac{B}{x+2}$ and solve for A and B.
• What is the first step for decomposing an improper fraction? Express it as the sum of a polynomial and a proper fraction.
• How do you handle repeated factors in partial fractions? Use terms like $\frac{A}{x+1} + \frac{B}{(x+1)^2}$.

What Examiners Usually Test

• Ability to decompose fractions with distinct linear factors.
• Correct handling of improper fractions before decomposition.
• Understanding of repeated factors in partial fraction decomposition.