Factorisation

1Summary 重點摘要

Split polynomial into product form, learn common factor extraction and cross method.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Extract the common factor firstThe first move in factorisation is always to look for a common factor in every term, covering both the numerical factor and the letter part, and take it out completely. For example, in 2x² + 4x every term contains 2 and x, so the expression becomes 2x(x + 2). Extracting first keeps the remaining expression simple and makes the cross method easier; forgetting this step, such as treating 2x² + 4x as if it were x² + 2x, gives an incomplete answer that loses marks.
2. Step 2: Recognise the quadratic trinomialA quadratic trinomial has the form x² + ax + b: three terms in total with highest degree two. The goal of factorisation is to write it as a product of two linear factors, (x + m)(x + n). Expanding (x + m)(x + n) gives x² + (m + n)x + mn, so comparing coefficients shows m + n = a and mn = b. Once this correspondence is understood, the cross method becomes reasoned deduction rather than a memorised chant, and any quadratic trinomial can be handled calmly.
3. Step 3: The principle of the cross methodThe cross method searches for two numbers m and n whose product equals the constant term b and whose sum equals the coefficient a. Both conditions are essential: the product corresponds to the mn part after expansion, and the sum corresponds to the (m + n)x part. Write the two conditions down before testing numbers; for x² + 5x + 6, note that the product is 6 and the sum is 5, and with a clear target the search becomes focused instead of blind guessing, saving valuable time.
4. Step 4: Test number pairs systematicallyList every integer pair whose product equals b, then check which pair has a sum equal to a. For b = 6 the pairs are (1, 6) and (2, 3) with sums 7 and 5, so 2 and 3 are chosen. Note that a negative b forces one positive and one negative number, while a positive b with a negative a forces both numbers to be negative. Listing the pairs in order avoids repetition and omission, and is the fastest reliable way to find the right numbers.
5. Step 5: Write the factorisationOnce m and n are found, write the original expression as (x + m)(x + n). For x² + 5x + 6, since 2 and 3 satisfy the conditions, x² + 5x + 6 = (x + 2)(x + 3). The two brackets may swap order because multiplication is commutative, but the signs must be exact: when m and n are both negative, the factors take the form (x - 2)(x - 3), and the signs must be written correctly together with the numbers.
6. Step 6: Check by expandingAfter factorising, expand the answer once to confirm it returns the original expression. For example (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6, exactly the original. The check takes only a few seconds yet catches almost every wrong pair or sign; if the expansion disagrees with the original, the middle coefficient usually fails to match, so return to the pair-finding step and recheck the sum instead of submitting an unchecked, possibly wrong answer.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Factorisation is opposite of expand
• Extract common factor first, then factor quadratic
• Cross method: find two numbers multiply = b, add = a
• Constant term positive and linear term negative, both numbers negative
• After factorise always expand back to check