Factorisation: Common Factors

1Summary 重點摘要

Factorisation turns a sum into a product by taking out the common factor from every term. For example, 6x² + 4x = 2x(3x + 2). When you're done, the bracket should have no common factor left.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What factorisation meansFactorisation rewrites a polynomial from a sum into a product, the exact reverse of expanding. For example, 6x² + 4x is a sum of two terms; factorised it becomes 2x(3x + 2), the product of 2x and (3x + 2). Expanding removes brackets while factorising creates them, and the two undo each other. Once mastered, factorisation makes solving equations and simplifying fractions far easier, because a product lets you apply the rule that if a product is zero, at least one factor is zero.
2. Step one: extract the number GCDFirst inspect the coefficients and find their greatest common factor. In 6x² + 4x the coefficients are 6 and 4: the factors of 6 are 1, 2, 3, 6 and those of 4 are 1, 2, 4, so the largest common one is 2, and the number part extracted is 2. Taking the greatest factor clears it in one move; extracting a smaller one leaves common factors inside, as in 2(3x² + 2x), which is unfinished and still needs an x pulled out.
3. Step two: extract the letter factorNext inspect the letters: for each variable take the lowest power that appears in every term. In 6x² + 4x, the first term holds x² (that is x × x) and the second holds x, so the common letter part is x, and together the common factor is 2x. A variable missing from any term cannot be extracted: in 6x² + 4 the second term has no x, so only the number 2 comes out, giving 2(3x² + 2).
4. Step three: divide every term by the factorWrite the common factor outside the bracket, then divide every term of the original expression by it and place the quotients inside. For 6x² + 4x: 6x² ÷ 2x = 3x and 4x ÷ 2x = 2, giving 2x(3x + 2). To verify, multiply back: 2x × 3x = 6x² and 2x × 2 = 4x, matching the original. Common errors are dropping a constant term by forgetting to divide it, or copying the exponent inside as in the mistaken 2x(3x² + 2).
5. Verify by expandingAfter factorising, always verify by expanding: multiply the common factor back through the bracket and the result must equal the original exactly. Expanding 2x(3x + 2) gives 6x² + 4x, confirming the work. This habit immediately exposes missing terms, wrong signs and wrong exponents. With the discipline of always checking, most factorisation errors in an exam are caught and corrected on the spot, and marks are saved.
6. A harder example and recurring mistakesTwo variables follow the same method: in 12a³b − 18a²b², the GCD of 12 and 18 is 6, the lowest power of a is a², and b appears in both terms, so the common factor is 6a²b. Dividing term by term gives 2a and 3b, hence 6a²b(2a − 3b). Three mistakes recur: extracting only the number and not the letters; taking the wrong lowest power such as a³; and, when a term is fully extracted, forgetting the 1 inside the bracket — 5x² + x factorises to x(5x + 1), not x(5x).

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Find the GREATEST common factor (number GCD × lowest power of each letter)
• Divide every term by it
• The minus sign travels with the term (−4x ÷ 2x = −2)
• Take the LOWEST power of each letter, not the highest
• Check: expanding must give back the original