Factorising Expressions
Factorise expressions by taking out the highest common factor of all terms, including shared letter factors, and check by expanding the brackets.
What a learner can do afterwards
- Identify the highest common factor of all terms in an expression
- Write the factorised form as a product of the HCF and a bracket
- Check factorisation by expanding the brackets back out
1 · Read
Factorising writes a sum or difference as a product. To take out the highest common factor (HCF), find factors shared by every term. For positive whole numbers, the HCF is the largest factor they share. List all factors and choose the largest in both lists. For 15 and 25, the HCF is 5. In 15x + 25, x is not shared because the constant term has no x.
Divide every term by the HCF. For 15x + 25, divide by 5: 15x ÷ 5 = 3x and 25 ÷ 5 = 5. Write 5(3x + 5). A coefficient equal to the HCF leaves 1: 5x ÷ 5 = x. Keep each term's sign. For 18x - 30, dividing by 6 gives 3x and -5, so the result is 6(3x - 5).
Check by expanding and comparing every term: 5(3x + 5) expands to 15x + 25. One matching substituted value does not prove equality for every value. An equivalent factorisation can be incomplete. If every term inside still shares a factor greater than 1, take it out too. If the numerical HCF is 1, no larger positive integer factor can be taken out.
With three terms, the HCF must divide all three coefficients. For 6a + 9b + 15, it is 3. Divide every term: 6a ÷ 3 = 2a, 9b ÷ 3 = 3b and 15 ÷ 3 = 5. The result is 3(2a + 3b + 5). No letter occurs in all three terms. Use the same method when a term is negative, keeping its sign.
Common factors can include letters. Write each term as a product and find factors present in every term. In 10z² + 15z, z² = z × z, so both terms contain z. Their coefficients share 5. Taking out 5z gives 5z(2z + 3). For 10zw + 15w, only w is shared by both terms, giving 5w(2z + 3). With subtraction, keep the sign of the quotient inside the bracket.
Find the factor shared by every term, including any shared letters. Write it outside a bracket, divide every term by it, then expand to check.
The lesson, questions and flashcards were written for this dataset.
All ten extra questions and all three readings were written for this dataset.
More to read
Finding a shared numerical factor
Start with complete factor lists when you need the highest common factor.
The factors of 20 are 1, 2, 4, 5, 10 and 20. The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Both lists contain 1, 2, 5 and 10. Choose 10, the largest shared factor. Greatest common factor, or GCF, is another name for HCF.
For 20p + 30, divide both terms by 10. The quotients are 2p and 3, giving 10(2p + 3). The constant 30 has no p, so p is not part of the common factor.
Multiply back to check: 10 × 2p = 20p and 10 × 3 = 30. Both original terms return.
When a letter is shared
Writing terms as products helps you spot shared letters.
For 14q² + 21q, write the terms as 7q × 2q and 7q × 3. Each contains 7q. Their remaining factors, 2q and 3, go inside the bracket: 7q(2q + 3).
Compare this with 14q + 21. Only its first term contains q. The common factor is 7, so the result is 7(2q + 3).
A factor outside a bracket must multiply every term inside. Taking a letter from only one original term would change the other term. Expand the completed bracket to check that each coefficient, letter and sign returns.
Equal expressions and complete factorisations
There are two checks to make after taking out a common factor.
First expand the bracket. Every resulting term must match the original expression. For example, 2(9m + 15) expands to 18m + 30, so those expressions are equal for every m.
Then inspect the terms inside. Here 9m and 15 still share 3. Taking out that 3 gives 2 × 3(3m + 5), which is 6(3m + 5). The full numerical HCF was 6.
Expansion checks equality. Looking for another factor shared by all terms inside checks whether the HCF has been taken out fully. Agreement at one substituted value cannot replace the expansion check.
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