Factorisation
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Common Factors and Factorising
To factorise means to write an expression as a product of factors. For 6x + 9, both terms share the factor 3, so it becomes 3(2x + 3). First write each term as a product of irreducible factors, find what is common, take it outside the bracket, and keep what is left inside.
Why it matters. Factorising makes expressions simpler and prepares children for solving equations. Many children stop at a partial factorisation or forget a common letter.
Try it in two minutes
- Ask your child to write 12 and 18 as products of prime numbers and find what they share.
- Now write 12x + 18 on paper and ask what number can be taken out. The answer is 6(2x + 3).
- Multiply back 6(2x + 3) and check that the original comes back.
- Try 5ab + 10a and ask if a letter is common too.
Try at home. Give your child 4x + 8, 9y - 6 and 10ab + 15a, and ask them to factorise each and check by expanding.
Factorisation by Regrouping
Sometimes no factor is common to all the terms, as in 2xy + 2y + 3x + 3. We group the terms in pairs: 2y(x + 1) + 3(x + 1). Now (x + 1) is common to both groups, so the answer is (x + 1)(2y + 3). Sometimes the terms must be rearranged to find a working grouping.
Why it matters. Regrouping trains children to look for patterns and try different options, which is a key problem-solving habit.
Try it in two minutes
- Write xy + 2y + 3x + 6 on paper and ask if any factor is common to all four terms.
- Group the first two terms and the last two terms and take out the common factors.
- Ask your child what bracket now appears in both groups.
- Multiply back to check the final answer.
Try at home. Ask your child to factorise ab + 3b + 2a + 6 and to check by expanding.
Factorisation Using Identities
Three identities help us factorise quickly: a² + 2ab + b² is (a + b)², a² - 2ab + b² is (a - b)², and a² - b² is (a + b)(a - b). To use them, check whether the first and last terms are squares and whether the middle term is twice their product.
Why it matters. Recognising these patterns makes algebra faster and is useful for simplifying fractions and solving equations later.
Try it in two minutes
- Ask your child to find 6 × 6, 5 × 5 and 7 × 7, so square numbers are fresh.
- Write x² + 10x + 25 and ask if 25 is a square and what 2 × x × 5 is.
- Write the answer as (x + 5)² and expand it to check.
- Write 49 - 25 and ask if it matches (7 - 5)(7 + 5).
Try at home. Ask your child to make three expressions of their own that fit the three identities and swap with you to factorise.
Factors of the Form (x + a)(x + b)
For x² + 5x + 6, we look for two numbers whose product is 6 and whose sum is 5. These are 2 and 3, so the factors are (x + 2) and (x + 3). The signs matter: a negative last term means one number is negative, and the larger one in size takes the sign of the middle term.
Why it matters. This is the most common type of factorisation in later classes and links directly to solving quadratic equations.
Try it in two minutes
- Write x² + 7x + 10 and ask for two numbers that multiply to 10 and add to 7.
- Let your child list pairs: 1 and 10, 2 and 5. Which pair adds to 7?
- Write the factors as (x + 2)(x + 5) and expand to check.
- Try x² - 3x - 10 and see how the signs change.
Try at home. Give two or three practice questions and ask your child to build the factors by listing number pairs.
Division of Algebraic Expressions
Division undoes multiplication. Since 2x times 3x² is 6x³, we know 6x³ ÷ 2x is 3x². To divide a polynomial by a monomial, divide each term, or take out the common factor and cancel. To divide by a polynomial, factorise first and cancel the common bracket. For example, (7x² + 14x) ÷ (x + 2) is 7x.
Why it matters. Dividing expressions is the reverse of multiplying and checks factorisation work. A common error is dividing only one term of the top.
Try it in two minutes
- Ask your child: 3 × 4 is 12, so what is 12 ÷ 3 and 12 ÷ 4? Link this to 2x × 3x which is 6x².
- Ask for 6x² ÷ 2x using the multiplication fact.
- Try (4x + 8) ÷ 4 and see that both terms are divided.
- Try (5x + 10) ÷ (x + 2) by factorising the top as 5(x + 2).
Try at home. Ask your child to make up a division from a multiplication, such as 4a × (a + 3), and then reverse it.
What your child will practise
- Common Factors and Factorising: Number Quest (Match It Up)
- Division of Algebraic Expressions: Candy Quick Fire (Quick Fire)
- Factorisation by Regrouping: Lab Explorer Quest (Collect the Answer)
- Factorisation Using Identities: Space Puzzle (Clue Puzzle)
- Factors of the Form (x + a)(x + b): Ocean Quick Fire (Quick Fire)
- Quiz: Common Factors and Factorising (8 questions)
- Quiz: Division of Algebraic Expressions (8 questions)
- Quiz: Factorisation Using Identities (8 questions)
- Quiz: Factorisation by Regrouping (8 questions)