Learning Objectives
- Factorise algebraic expressions using common factors
- Factorise by regrouping terms
- Factorise using algebraic identities
- Divide algebraic expressions (polynomial by monomial and polynomial by polynomial)
Key Concepts
What is Factorisation?
Factorisation is the process of writing an algebraic expression as a product of its factors. It is the reverse of expansion (multiplication). For example: 6x + 12 = 6(x + 2).
Method 1: Common Factor Method
Find the HCF (Highest Common Factor) of all terms and take it out as a common factor.
Example: 12x²y + 18xy² = 6xy(2x + 3y). Here, 6xy is the HCF of 12x²y and 18xy².
Method 2: Regrouping
When there is no single common factor for all terms, rearrange and group terms to find common factors within groups.
Example: ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y).
Method 3: Using Identities
Recognise expressions that match standard identities and factorise accordingly:
- a² + 2ab + b² = (a + b)²
- a² - 2ab + b² = (a - b)²
- a² - b² = (a + b)(a - b)
- x² + (a+b)x + ab = (x + a)(x + b)
Example: 9x² - 16 = (3x)² - 4² = (3x + 4)(3x - 4).
Example: x² + 10x + 25 = x² + 2(5)(x) + 5² = (x + 5)².
Division of Algebraic Expressions
Monomial ÷ Monomial: Divide coefficients and subtract exponents of like variables.
Polynomial ÷ Monomial: Divide each term of the polynomial by the monomial separately.
Polynomial ÷ Polynomial: Factorise both the dividend and divisor, then cancel common factors.
Summary
Factorisation is the reverse of multiplication. The three main methods are: taking out common factors, regrouping, and using standard algebraic identities. Division of algebraic expressions relies on factorisation to simplify. Always check by expanding the factors to verify the result.
Important Terms
- Factor: A number or expression that divides another exactly
- HCF: Highest Common Factor of the terms
- Factorisation: Expressing an expression as a product of factors
- Identity: An equation true for all variable values
- Irreducible Factor: A factor that cannot be factorised further
Quick Revision
- Common factor method: find HCF of all terms and factor it out
- Regrouping: rearrange terms into groups with common factors
- a² + 2ab + b² = (a + b)²; a² - 2ab + b² = (a - b)²
- a² - b² = (a + b)(a - b)
- Division: factorise both expressions, then cancel common factors
- Always verify by expanding the factorised form