Factorisation
Overview
Factorisation is the process of writing an algebraic expression as a product of its factors. Just as numbers can be expressed as products of prime factors, algebraic expressions can be written as products of their irreducible factors. This chapter covers methods like finding common factors, grouping, using standard identities, and division of algebraic expressions. Understanding factorisation helps simplify complex expressions and solve equations efficiently.
Typed Notes
Method 1 – Common Factor Method
In this method, we find the HCF of all the terms of the expression and take it outside as a common factor. For example, 12x² + 6x = 6x(2x + 1). Here 6x is the HCF of both terms. Each term is then written as a product of the HCF and the remaining factor. This is the simplest and most basic method of factorisation. Always check first whether a common factor exists before trying other methods.
Method 2 – Factorisation by Regrouping
Sometimes no single factor is common to all terms, but we can rearrange and group terms so that each group shares a common factor. For example, 2xy + 2y + 3x + 3 = 2y(x+1) + 3(x+1) = (x+1)(2y+3). The key step is identifying how to group the terms. After grouping, a common binomial factor appears, which is then taken out. This method requires practice in recognising suitable groupings.
Method 3 – Using Standard Identities
Several expressions can be factorised by recognising them as standard algebraic identities. The main identities used are: (a+b)² = a²+2ab+b², (a-b)² = a²-2ab+b², and (a+b)(a-b) = a²-b². For example, x²+6x+9 = (x+3)², and 25x²-16 = (5x+4)(5x-4). We compare the given expression with the identity, identify a and b, and write the factorised form directly. Recognising the pattern is the most important skill here.
Division of Algebraic Expressions
Division of an algebraic expression is the reverse of multiplication. To divide a polynomial by a monomial, divide each term of the polynomial separately by the monomial. For example, (6x²+4x) ÷ 2x = 3x+2. To divide a polynomial by a polynomial, factorise both the dividend and divisor, then cancel the common factors. For example, (x²+5x+6) ÷ (x+2) = (x+2)(x+3) ÷ (x+2) = (x+3). Always factorise before dividing.
Key Concepts
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- 01Factorisation means expressing an expression as a product of two or more factors that cannot be further simplified.
- 02Common factor method: Take out the highest common factor (HCF) from all terms of the expression.
- 03Regrouping method: Group terms suitably so that a common factor emerges from each group.
- 04Identity method uses standard algebraic identities such as (a+b)², (a-b)², and (a+b)(a-b) to factorise expressions.
- 05Division of a polynomial by a monomial or polynomial is done by factorising both and then cancelling common factors.
Important Formulas
Questions & Solutions
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Factorise: 18x²y + 12xy²
Factorise: x² + 8x + 16
Factorise: 49y² - 36
Factorise by regrouping: 6ab + 9b + 4a + 6
Divide: (x² + 7x + 12) ÷ (x + 3)
Factorisation expresses an algebraic expression as a product of its factors using four main methods: common factor, regrouping, standard identities, and division. The standard identities (a+b)², (a-b)², and (a²-b²) are especially useful. Division of algebraic expressions is simplified by factorising both numerator and denominator and cancelling common factors. Mastering these methods builds a strong foundation for higher algebra.
Chapter-end quiz
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Q1.Which of the following is the correct factorisation of x² - 25?
Q2.What is the HCF of 15x³y and 10x²y²?
Q3.Factorise: a² - 6a + 9
Q4.What is (4x² + 8x) ÷ 4x?
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