Factorisation – JSS3 Mathematics Lesson Note (Third Term)

Factorisation is the process of writing an expression as a product of its factors.

Behavioural objectives

By the end of the lesson, students should be able to:

1. factorise simple expressions by taking out the HCF;
2. state the meaning of a quadratic expression, a quadratic equation and a trinomial;
3. factorise trinomials of the form x² + bx + c by grouping;
4. factorise quadratic expressions of the form ax² + bx + c;
5. factorise the difference of two squares.

What are the key terms used in factorisation?

These are the words students must know before they start factorising.

• Factor: a number or expression that divides another exactly, for example 3 and x are factors of 3x.
• Term: a part of an expression separated by + or – signs, for example 3x and 5 in 3x + 5.
• Coefficient: the number multiplying a letter, for example 3 in 3x.
• Constant: a term with no letter, for example 6 in x² + 7x + 6.
• HCF: the highest common factor, the largest factor shared by all the terms.
• Expanding: removing brackets by multiplying, which is the opposite of factorising.

How do we factorise a simple expression?

To factorise an expression completely, take the HCF outside the bracket and divide each term by the HCF.

Steps

1. Find the HCF of all the terms.
2. Write the HCF outside a bracket.
3. Divide each term by the HCF and write the results inside the bracket.
4. Expand your answer to check that it gives the original expression.

Examples

1. 3x + 6: the HCF is 3, so 3x + 6 = 3(x + 2).
2. 4x² + 8x: the HCF is 4x, so 4x² + 8x = 4x(x + 2).
3. x(x + 6) + 1(x + 6): the HCF is the bracket (x + 6), so the answer is (x + 6)(x + 1).

What is a quadratic expression?

A quadratic expression is an expression whose highest power is two (2), so it is sometimes called a second-order polynomial.

• General form: ax² + bx + c, where a ≠ 0 and a, b and c stand for numbers.
• Examples: x² + 7x + 6 and 5x² – 9x + 4.

Points to note

1. Quadratic equation: if ax² + bx + c = 0, it is known as a quadratic equation.
2. Coefficients: a is the coefficient of x², b is the coefficient of x and c is a constant term.
3. Trinomial: an expression with three terms is known as a trinomial.
4. Four terms: to factorise a trinomial by grouping, we first convert it to contain four terms.

How do we factorise a trinomial of the form x² + bx + c?

A trinomial of this form is factorised by splitting the middle term into two terms and then grouping.

Steps

1. Multiply the 1st term and the last term (3rd term) of the expression.
2. Find two factors of this product that add up to the second (middle) term and multiply to give the result in step 1.
3. Replace the middle term with these two numbers and factorise by grouping.

Worked example 1: factorise x² + 7x + 6

1. Product of 1st and last terms: x² × 6 = 6x²
2. Factors that add up to 7x: 6x and x
3. Split the middle term: x² + 6x + x + 6
4. Group: x(x + 6) + 1(x + 6)
5. Answer: (x + 6)(x + 1)

Worked example 2: factorise x² – 2x – 15

1. Product: x² × (-15) = -15x²
2. Factors that multiply to -15 and add up to -2: -5 and +3
3. Split the middle term: x² – 5x + 3x – 15
4. Group: x(x – 5) + 3(x – 5)
5. Answer: (x – 5)(x + 3)

How do the signs help us choose the two numbers?

How do we factorise a quadratic expression of the form ax² + bx + c?

A quadratic expression with a coefficient of x² greater than 1 is factorised with the same steps, using the product a × c.

Worked example 1: factorise 5x² – 9x + 4

1. Product: 5x² × 4 = 20x²
2. Factors: -5 and -4
3. Check the sum: -5 – 4 = -9
4. Split the middle term: 5x² – 5x – 4x + 4
5. Group: 5x(x – 1) – 4(x – 1)
6. Answer: (5x – 4)(x – 1)

Worked example 2: factorise 2x² + 7x + 3

1. Product: 2x² × 3 = 6x²
2. Factors that multiply to 6 and add up to 7: 6 and 1
3. Split the middle term: 2x² + 6x + x + 3
4. Group: 2x(x + 3) + 1(x + 3)
5. Answer: (2x + 1)(x + 3)

How do we factorise the difference of two squares?

The difference of two squares is an expression of the form x² – y², and it factorises as (x + y)(x – y).

Steps

1. Write each term as a perfect square.
2. Find the square root of each term.
3. Write two brackets: one with + between the roots and one with – between them.

Examples

1. P² – Q² = (P + Q)(P – Q)
2. 36y² – 1 = 6²y² – 1²
   = (6y)² – 1²
   = (6y + 1)(6y – 1)
3. 49 – x² = 7² – x² = (7 + x)(7 – x)
4. 25a² – 16b² = (5a)² – (4b)² = (5a + 4b)(5a – 4b)

What mistakes should students avoid in factorisation?

Most errors come from wrong signs or from stopping before the factorisation is complete.

• Not taking the full HCF: always check that nothing common is left inside the bracket.
• Sign errors: the two numbers must multiply to give the product and add up to the middle term.
• A sum of two squares: x² + y² does not factorise into two brackets like the difference of two squares.
• Skipping the check: expand the brackets to confirm the answer matches the original expression.

Evaluation

Factorise each of the following:

1. x² + 10x – 24
2. 2x² + 13x + 6
3. 13d² – 11d – 2
4. 121 – y²
5. x²y² – 4²

Assignment

Reading assignment

• Essential Mathematics for J.S.S.3, pages 29-36
• Exam Focus for J.S.S.C.E., pages 101-105

Weekend assignment

1. The coefficient of x² in x² + 3x – 5 is (a) 3 (b) 1 (c) -5
2. Factorise e² – f²: (a) (e + f)(e – f) (b) (e + f)(f + e) (c) (e – f)(f – e)
3. Factorise x² + x – 6: (a) (x + 3)(x + 2) (b) (x – 2)(x + 3) (c) (x + 1)(x + 5)
4. Factorise by grouping 5h² – 20h + h – 4: (a) (h – 4)(5h + 1) (b) (h + 4)(5h – 1) (c) (h + 2)(h – 5)
5. 49m² – 64n² when factorised will be: (a) (7m + 8n)(8m + 7n) (b) (8m – 7n)(8m + 7n) (c) (7m – 8n)(7m + 8n)

Theory

Factorise the following:

1. 4p² – 12p + 9
2. f² – 2f + 1