Factorisation — Study Notes

1. Introduction

When we break down a number into its factors, we express it as a product of smaller numbers. For example, 12 = 2 x 2 x 3. In the same way, algebraic expressions can also be broken down into their factors. The process of expressing an algebraic expression as a product of its factors is called factorisation. Factorisation is the reverse of multiplication: while multiplication combines factors into an expression, factorisation breaks the expression back into factors.

In this chapter we will learn different methods of factorisation: taking out common factors, grouping terms, using the standard identities, and splitting the middle term of quadratic expressions. We will also learn how to divide algebraic expressions using the techniques of cancellation and the division of monomials and polynomials. Factorisation is an essential skill that is used in solving equations, simplifying expressions and in all higher mathematics.

2. Factorisation by Common Factors

When all the terms of an expression share a common factor, we can take it out. This is called the method of common factors, and it is based on the distributive law ab + ac = a(b + c).

Steps

1. Find the common factor of all terms.
2. Take it outside the brackets.
3. Divide each term by the common factor and write the quotient inside the brackets.

Example

Factorise 6x^2 + 12x. The common factor is 6x. So 6x^2 + 12x = 6x(x + 2).

3. Factorisation by Grouping

When an expression has four or more terms and there is no common factor for all of them, we group the terms such that each group has a common factor.

Example

Factorise ab + bc + ad + cd. Group as (ab + bc) + (ad + cd) = b(a + c) + d(a + c) = (a + c)(b + d).

Example with Sign Care

Factorise a^2 - ab - ac + bc. (a^2 - ab) - (ac - bc) = a(a - b) - c(a - b) = (a - b)(a - c).

4. Factorisation using Identities

The standard identities can be used in reverse to factorise expressions.

Identity I (in reverse)

$$a^2 + 2ab + b^2 = (a + b)^2$$

Example: x^2 + 6x + 9 = (x + 3)^2.

Identity II (in reverse)

$$a^2 - 2ab + b^2 = (a - b)^2$$

Example: x^2 - 8x + 16 = (x - 4)^2.

Identity III (in reverse)

$$a^2 - b^2 = (a + b)(a - b)$$

Example: 25x^2 - 9 = (5x + 3)(5x - 3).

5. Splitting the Middle Term

To factorise a quadratic of the form x^2 + (a + b)x + ab, we find two numbers p and q such that p + q equals the coefficient of x and p x q equals the constant term. Then we split the middle term.

Example

Factorise x^2 + 5x + 6. We need two numbers whose sum is 5 and product is 6: they are 2 and 3. x^2 + 5x + 6 = x^2 + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 2)(x + 3).

6. Division of Algebraic Expressions

Division of a Monomial by a Monomial

Divide the coefficients and divide the variables using the laws of exponents. $$14x^4 \div 7x^2 = 2x^2$$

Division of a Polynomial by a Monomial

Divide each term of the polynomial by the monomial.

Division of a Polynomial by a Polynomial

Factorise both the numerator and the denominator, then cancel the common factors.

Example

$$\frac{7x^2 + 14x}{7x} = \frac{7x(x + 2)}{7x} = x + 2$$

Quick Revision Tables

Table 1: Methods of Factorisation

Table 2: Division Rules

Mind Map

Important Diagrams (SVG)

Diagram 1: Factorisation as the Reverse of Multiplication

Diagram 2: Splitting the Middle Term

Solved Example Approach

Consider the problem of factorising the expression 4x^2 - 9y^2. At first glance, this expression has two terms and no common factor, so neither the common factor method nor grouping will help directly. The key is to recognise the pattern of a difference of two squares. Since 4x^2 can be written as (2x)^2 and 9y^2 can be written as (3y)^2, the expression becomes (2x)^2 - (3y)^2. Applying Identity III in reverse, we get (2x + 3y)(2x - 3y). To verify the answer, we multiply the two factors back: (2x + 3y)(2x - 3y) = 4x^2 - 6xy + 6xy - 9y^2 = 4x^2 - 9y^2, which matches the original expression. This check confirms that our factorisation is correct, and it illustrates the general rule that verification by multiplication is always a wise final step.

Now consider a quadratic that requires splitting the middle term, such as x^2 - 7x + 12. We need two numbers whose sum is -7 and whose product is 12. The pairs that multiply to 12 are (1, 12), (2, 6) and (3, 4), and their sums are 13, 8 and 7 respectively. To obtain a sum of -7, both numbers must be negative, so we take -3 and -4, since (-3) + (-4) = -7 and (-3) x (-4) = 12. Splitting the middle term gives x^2 - 3x - 4x + 12. Grouping the first two and the last two terms, we get x(x - 3) - 4(x - 3), which equals (x - 3)(x - 4). Again we can check by multiplying: (x - 3)(x - 4) = x^2 - 4x - 3x + 12 = x^2 - 7x + 12. This approach works whenever the coefficient of x^2 is 1, and with practice the choice of the correct pair of numbers becomes quick and natural.

Common Mistakes

1. Students forget to take the negative sign along with the common factor. For example, -2x + 6 = -2(x - 3), not -2(x + 3).
2. While grouping, students change signs incorrectly, especially when a group starts with a minus sign.
3. Students apply identity III (difference of squares) to expressions that are not differences of squares, like x^2 + 9.
4. In splitting the middle term, students fail to find the correct pair of numbers because they do not check both the sum and the product conditions.
5. Students forget to divide each term by the common factor when factorising by the common factor method.
6. In division, students cancel terms instead of factors. Cancellation is allowed only for common factors, not common terms in sums.
7. Students forget that a^2 - b^2 = (a + b)(a - b) while wrongly writing it as (a - b)^2.

Exam Tips

1. Always check whether a common factor exists before trying other methods of factorisation.
2. Write each step of the factorisation clearly; partial method credit is often awarded.
3. Verify the final factorised form by multiplying the factors to get the original expression.
4. For difference of squares, recognise the pattern a^2 - b^2 and write the two factors immediately.
5. When dividing expressions, always factorise completely and then cancel common factors, not terms.
6. Practise splitting the middle term with many quadratics, since it appears frequently in exams.
7. Remember that the coefficient of x^2 must be 1 for the simple splitting method; otherwise, handle it with extra care.

Conclusion

Factorisation is one of the most important tools of algebra. In this chapter we learnt how to factorise expressions by taking out common factors, by grouping terms, by applying the standard identities in reverse, and by splitting the middle term of quadratics. We also studied how to divide algebraic expressions by factorising and cancelling common factors. These techniques are used everywhere in mathematics - in solving equations, simplifying rational expressions, and in working with polynomials in higher classes. Careful practice of factorisation builds algebraic fluency and confidence that will benefit students throughout their mathematical journey.