Algebra

Study Notes

In further algebra, students learn about improper algebraic fractions, partial fractions, and binomial expansions for non-positive integers. These concepts are essential for understanding complex algebraic expressions and their simplifications.

• Improper Algebraic Fractions — Fractions where the numerator's degree is greater than or equal to the denominator's degree. Example: $\frac{f(x)}{g(x)}$ where $\deg(f(x)) \geq \deg(g(x))$.
• Partial Fractions — A method to express a single fraction as a sum of simpler fractions. Example: $\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$.
• Binomial Expansion of (1 + x)ⁿ — Expanding expressions where n is a rational number and |x| < 1. Example: $(1 + x)^{-4} = 1 - 4x + 10x^2 - 20x^3 + \ldots$
• Binomial Expansion of (a + x)ⁿ — Expanding expressions with a factor outside the bracket for non-positive n. Example: $(2 + x)^{-3} = \frac{1}{8} - \frac{3x}{16} + \frac{15x^2}{64} + \ldots$

Exam Tips

Key Definitions to Remember

• Improper algebraic fraction: numerator's degree ≥ denominator's degree
• Partial fraction: expressing a fraction as a sum of simpler fractions
• Binomial expansion: expanding expressions using binomial theorem

Common Confusions

• Forgetting to divide improper fractions before partial fraction decomposition
• Misapplying binomial expansion for values of n that are not positive integers

Typical Exam Questions

• How do you express $\frac{3x^2 + 5x + 2}{(x+1)(x+2)}$ in partial fractions? Divide first, then decompose into partial fractions
• What is the binomial expansion of $(1 + x)^{-2}$ up to the x² term? $1 - 2x + 3x^2$
• How do you find the first 3 terms of $(3 + x)^{-1}$? $\frac{1}{3} - \frac{x}{9} + \frac{x^2}{27}$

What Examiners Usually Test

• Ability to decompose fractions into partial fractions
• Correct application of binomial expansion for non-positive integer n
• Understanding the conditions for convergence in binomial series