Algebraic Manipulation (Copy)

Algebraic Expressions Cheat Sheet (IGCSE Mathematics 0580)
Topic: Simplify, Expand, and Factorise

1. Simplifying Expressions by Collecting Like Terms

• Like terms are terms that have the same variables raised to the same powers.
• Only like terms can be added or subtracted.
• Examples:
  • 2a + 5a = 7a
  • 3x² – 5x² = –2x²
  • 4m + 7n – 2m + 5n = (4m – 2m) + (7n + 5n) = 2m + 12n
  • 2a + 3b + 5a – 9b = 7a – 6b
• Key Tip: Always arrange similar terms together before combining.

2. Expanding Products of Algebraic Expressions

a. Expand Single Bracket Expressions

• Multiply everything inside the bracket by what’s outside.
• Example:
  • Expand 3x(2x – 4y)
    → 3x × 2x = 6x²
    → 3x × (–4y) = –12xy
    → Final Answer: 6x² – 12xy

b. Expand Two Brackets (Binomial Expansion)

• Each term in the first bracket multiplies each term in the second bracket.
• Use FOIL (First, Outside, Inside, Last) method.
• Example:
  • Expand (2x + 1)(x – 4)
    → First: 2x × x = 2x²
    → Outside: 2x × (–4) = –8x
    → Inside: 1 × x = +x
    → Last: 1 × (–4) = –4
    → Combine: 2x² – 8x + x – 4 = 2x² – 7x – 4
• Another Example:
  • Expand (x + 3)(x + 5)
    → x² + 5x + 3x + 15 = x² + 8x + 15

3. Factorising Algebraic Expressions

• Factorising means writing an expression as a product of its factors.

a. Extracting Common Factors

• Find the highest common factor (HCF) from each term and factor it out.
• Examples:
  • Factorise 9x² + 15xy
    → HCF = 3x
    → 3x(3x + 5y)
  • Factorise 6a² – 12a
    → HCF = 6a
    → 6a(a – 2)

b. Factorising Simple Quadratics

• Quadratic form: ax² + bx + c
• Find two numbers that multiply to a×c and add to b.
• Example:
  • Factorise x² + 7x + 12
    • Numbers: 3 and 4 → 3×4 = 12, 3+4 = 7
    • (x + 3)(x + 4)
• Example:
  • Factorise x² – 5x + 6
    • Numbers: –2 and –3 → (x – 2)(x – 3)

Key Vocabulary

Important Shortcuts

• a(b + c) = ab + ac (distributive property)
• (a + b)(c + d) = ac + ad + bc + bd
• x² + (a + b)x + ab = (x + a)(x + b) (for simple quadratics)

Exam Tips

• Always expand first before simplifying, unless told to factorise.
• After factorising, you can expand back to check if the factorisation is correct.
• In factorising, always take out negative signs if the first term is negative (e.g., –3x(–2x + 5)).

This is the foundation for most algebra topics like solving equations, working with inequalities, and manipulating formulas.