--- title: "Factorising Expressions" description: "Factorise expressions by taking out the highest common factor of all terms, including shared letter factors, and check by expanding the brackets." canonical: https://lightmysky.com/learn/mathematics/factorising-expressions-mt_uESbzWCZIq source: https://lightmysky.com/learn/mathematics/factorising-expressions-mt_uESbzWCZIq.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Factorising Expressions Factorise expressions by taking out the highest common factor of all terms, including shared letter factors, and check by expanding the brackets. Subject: Mathematics · Area: Algebra · Ages 12 to 13 Page: https://lightmysky.com/learn/mathematics/factorising-expressions-mt_uESbzWCZIq ## Ready when they can - 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 ## Lesson: Factorising by Taking Out a Common Factor 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. **Example.** 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). *(drawing: A common factor must divide every coefficient. Complete lists show which shared factor is largest.)* 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. **Example.** 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. **Recap.** 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.* ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Expanding Single Brackets](https://lightmysky.com/learn/mathematics/expanding-single-brackets-mt_FX4a2Q8XXN) - [Sign Rules for Multiplication and Division](https://lightmysky.com/learn/mathematics/sign-rules-for-multiplication-and-division-mt_rxInpOQ74w) ## Opens up - [Factorising Quadratics into Two Brackets](https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA)