Algebraic Manipulation
2.2 Algebraic Manipulation – Cheat Sheet
1. Simplifying Expressions (Collecting Like Terms)
- Combine terms with exactly the same variables and powers.
- Add/subtract their coefficients.
| Example | Working | Answer |
|---|---|---|
| 2a² + 3ab − 1 + 5a² − 9ab + 4 | (2a² + 5a²) + (3ab − 9ab) + (−1 + 4) | 7a² − 6ab + 3 |
| 4x + 7y − 3x + 2y | (4x − 3x) + (7y + 2y) | x + 9y |
2. Expanding Products
Single Bracket: Multiply each term inside the bracket by the term outside.
- Example: 3x(2x − 4y) = 6x² − 12xy
Double Bracket (FOIL method): First, Outer, Inner, Last.
- Example: (3x + y)(x − 4y)
- First: 3x × x = 3x²
- Outer: 3x × (−4y) = −12xy
- Inner: y × x = xy
- Last: y × (−4y) = −4y²
- Result: 3x² − 11xy − 4y²
More than Two Brackets: Expand step-by-step.
- Example: (x − 2)(x + 3)(2x + 1)
- Step 1: (x − 2)(x + 3) = x² + 3x − 2x − 6 = x² + x − 6
- Step 2: (x² + x − 6)(2x + 1) = 2x³ + x² + 2x² + x − 12x − 6
- Combine: 2x³ + 3x² − 11x − 6
3. Factorising by Extracting Common Factors
- Take out the highest common factor (HCF) from all terms.
| Example | Working | Answer |
|---|---|---|
| 9x² + 15xy | 3x(3x + 5y) | 3x(3x + 5y) |
| 12a²b − 18ab² | 6ab(2a − 3b) | 6ab(2a − 3b) |
4. Factorising Special Forms
| Type | Example | Factorised Form |
|---|---|---|
| Common pattern ax + bx + kay + kby | ax + bx + kay + kby = x(a + b) + ky(a + b) | (x + ky)(a + b) |
| Difference of squares a²x² − b²y² | a²x² − b²y² = (ax − by)(ax + by) | (ax − by)(ax + by) |
| Perfect square trinomial a² + 2ab + b² | a² + 2ab + b² = (a + b)² | (a + b)² |
| Quadratic ax² + bx + c | x² + 5x + 6 = (x + 2)(x + 3) | Factor pairs of c sum to b |
| Cubic factor ax³ + bx² + cx | x(a x² + b x + c) | Take out x first, then factor quadratic |
5. Completing the Square (ax² + bx + c)
If a = 1:
- Take half of b, square it, and add/subtract inside.
- Example: x² + 6x + 5
- Half of 6 = 3, square = 9
- x² + 6x + 9 − 9 + 5 = (x + 3)² − 4
- Example: x² + 6x + 5
If a ≠1:
- Factor out a from x² and x terms before completing the square.
- Example: 2x² + 8x + 3
- Factor out 2: 2(x² + 4x) + 3
- Half of 4 = 2, square = 4
- 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5
- Example: 2x² + 8x + 3
6. Examples Table for Quick Reference
| Problem | Process | Answer |
|---|---|---|
| 2a² + 3ab − 1 + 5a² − 9ab + 4 | Collect like terms | 7a² − 6ab + 3 |
| Expand 3x(2x − 4y) | Multiply each term | 6x² − 12xy |
| Expand (3x + y)(x − 4y) | FOIL | 3x² − 11xy − 4y² |
| Factorise 9x² + 15xy | Take out HCF 3x | 3x(3x + 5y) |
| Factorise a²x² − b²y² | Difference of squares | (ax − by)(ax + by) |
| Complete square: x² + 6x + 5 | Half b = 3, square = 9 | (x + 3)² − 4 |