Polynomials and factorisation
How to expand and factorise polynomials: common factors, identities, grouping and the cross method. It is the most important algebra skill before F4, because quadratic equations, polynomials and algebraic fractions all need it.
What you will learn
- 11.1 Terms, monomials, binomials, degree, coefficients, constant terms, like terms; ascending and descending order.
- 11.2 Add, subtract and multiply polynomials, including more than one variable.
- 11.3 Factorise: common factors, grouping of terms, cross-method.
- 12.1 Identities versus equations; prove simple identities.
- 12.2–12.3 Expand and factorise with:
You need first: Number and algebra basics, Indices and scientific notation
Key terms
| English | 中文 |
|---|---|
| polynomial, term | 多項式，項 |
| monomial, binomial | 單項式，二項式 |
| degree | 次數 |
| coefficient, constant term | 係數，常數項 |
| like terms | 同類項 |
| expand / factorise | 展開／因式分解 |
| common factor | 公因式 |
| grouping terms | 分組分解 |
| cross-method | 十字相乘法 |
| identity | 恆等式 |
| difference of two squares | 平方差 |
| perfect square | 完全平方 |
Standard steps
The order to try (every time)
- Common factor first. Take out the H.C.F. of all terms, including a negative sign if the first term is negative.
- Count the terms:
- 2 terms: is it a difference of two squares, a² − b²?
- 3 terms: is it a perfect square? If not, use the cross-method.
- 4 terms: group into pairs, factorise each pair, then take out the common bracket.
- Check each bracket to see if it can be factorised again.
Cross-method for ax² + bx + c
- List factor pairs of a (left column) and of c (right column).
- Cross-multiply and add. The sum must be the middle coefficient b.
- Read the brackets across the rows.
2x +3 2x × (−5) = −10x
╳ 3x × (+3) = +9x
3x −5 −x ✓ matches −xSo 6x² − x − 15 = (2x + 3)(3x − 5).
Tip on signs: if c is positive, both signs are the same as b's sign. If c is negative, the signs are different.
Prove an identity
- Start from one side, usually the more complicated one.
- Expand and simplify until it equals the other side.
- End with "∴ L.H.S. = R.H.S." Never move terms across the "≡".
Worked examples
Cross-method above: (2x + 3)(3x − 5).
= a(x − y) + 2(x − y) group in pairs
= (x − y)(a + 2) common bracket(DSE style: two methods in one.)
= (x + y)(x − y) + 3(x − y)
= (x − y)(x + y + 3)9x² = (3x)², 16 = 4², 2(3x)(4) = 24x, so it is (3x − 4)².
= (2m)² − (5n)² = (2m + 5n)(2m − 5n).
3x −2 3x × (+4) = +12x
╳ x × (−2) = −2x
x +4 +10x ✓ matches +10xc is negative, so the signs differ. 3x² + 10x − 8 = (3x − 2)(x + 4).
Common mistakes
- Forgetting the common factor first, so the answer is not "fully factorised".
- Sign errors in grouping: −ay − 2y must become −y(a + 2) with the minus outside.
- (a − b)² expanded as a² − b².
- Treating a² + b² as a difference of squares (it doesn't factorise over real numbers).
- Stopping early: x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2).
How the DSE tests it
- A(1): a 3–4 mark question with two parts, where part (b) uses part (a)'s factor. For example, factorise a quadratic in (a), then use that bracket in a four-term grouping in (b).
- Algebraic fractions: factorise each denominator before combining.
- Paper 2: quick factorisation or expansion items.
Questions you might not ask in class
How do I guess the right numbers in the cross-method?
It's organised trial, not guessing. Start with the factor pairs closest together. With practice, you check 2–3 pairs, not all of them.
How do I know I've finished?
Expand your answer to check it, then look at each bracket: can it be factorised again?
What is the difference between = and ≡?
An equation is true for some values of x. An identity is true for every value of x.