Summary
The binomial expansion allows us to expand expressions of the form (1 + x)ⁿ and (a + bx)ⁿ, where n is a rational number and certain conditions are met. Partial fractions are used to express rational functions in a simpler form for expansion.
- Binomial Expansion (1 + x)ⁿ — Expanding (1 + x)ⁿ is valid for |x| < 1 and n as a rational number. Example: For n = -4, the series is infinite and converges when |x| < 1.
- Binomial Expansion (a + bx)ⁿ — To expand (a + bx)ⁿ, factor out 'a' and ensure |x/a| < 1. Example: Find the first 3 terms of (2 + 3x)ⁿ.
- Partial Fractions — Decompose rational functions into simpler fractions for expansion. Example: Express (ax+b)/(cx+d) in partial fractions and expand.
Exam Tips
Key Definitions to Remember
- Binomial expansion for (1 + x)ⁿ
- Binomial expansion for (a + bx)ⁿ
- Partial fraction decomposition
Common Confusions
- Forgetting that the binomial series is infinite when n is not a positive integer
- Misapplying the condition |x| < 1 for convergence
Typical Exam Questions
- How do you expand (1 + x)ⁿ for n = -2? Use the binomial theorem with |x| < 1.
- How do you express (3x+2)/(x²+5x+6) in partial fractions? Decompose into simpler fractions.
- What are the first 3 terms of (2 + 3x)ⁿ? Factor out '2' and expand using the binomial theorem.
What Examiners Usually Test
- Ability to apply the binomial theorem for non-integer n
- Correct use of partial fractions in expansions
- Understanding of convergence conditions for binomial series