Algebraic Factorisation (Complete Lesson)
High Contrast Educational Guide • Rules, Applications & Practice
Exercises Navigation
4 Exercises1 What is Factorisation? (Concept & Relationships)
A) Algebraic Relationship
Expansion and Factorisation are inverse operations:
- • Expansion: (x + 2)(x + 3) → x² + 5x + 6
- • Factorisation: x² + 5x + 6 → (x + 2)(x + 3)
Factoring x² + 5x + 6 = 0 gives roots x = -2, -3.
B) Geometric Relationship
Geometrically, factorisation relates to Area & Dimensions:
If Area = x² + 5x + 6, then Length = x + 3 and Width = x + 2.
2 4 Main Rules of Factorisation (Step-by-Step)
Check for common numbers or variables across all terms:
3x² + 6x = 3x(x + 2)
ax + ay + bx + by = (x + y)(a + b)
Apply standard algebraic expansion formulas:
For three-term perfect square trinomials:
For quadratic expressions ax² + bx + c:
3 Real-World Applications
1. Security & Cryptography
Digital banking and encryption (RSA algorithm) rely on prime factorisation of huge numbers.
2. Architectural Engineering
Structural stress and balance equations for bridges and buildings are simplified using factorisation.
3. Physics & Kinematics
Projectile motion, flight trajectories, and impact points are solved via quadratic factorisation.
4. Economics & Optimization
Used to calculate maximum profit margins, break-even points, and cost optimization.
Tips for Solving
- • Area Model: Visualize \(x^2\) as square and \(x\) as rectangular strip.
- • Sign Rules: A minus sign outside brackets flips all internal signs.
- • Complete Check: Always check if inner expressions can be factored further.
Common Mistakes
- • Skipping Common First: On 2x² – 8, factor out 2 first: 2(x – 2)(x + 2).
- • Bracket Sign Error: Note that -(x – 3) = -x + 3.
- • Incomplete Factorisation: Don’t stop at (x² – 4)(x² + 4).
Quick Practice Quiz
Question: What is the correct factorisation of x² – 25?