Why expansions and factorisation matter
Ever wondered how a messy polynomial turns into a neat product? That trick is the heart of many ICSE algebra questions. Get it right, and the exam feels a lot less scary.
In simple words, expanding means turning a product like (x+2)(x‑3) into a single polynomial, while factorising is the reverse – breaking a polynomial into smaller pieces that multiply back to the original.
Expansions made easy
Think of expansion like spreading butter on toast. You start with a compact block (the brackets) and spread it out into a smooth layer (the polynomial).
Basic binomial expansion
A binomial is an expression with exactly two terms, like (a+b). When you multiply two binomials, use the FOIL method – First, Outer, Inner, Last. It’s just a fancy way to remember which terms to pair.
- First: multiply the first terms of each binomial.
- Outer: multiply the outermost terms.
- Inner: multiply the innermost terms.
- Last: multiply the last terms.
Example:
(x+2)(x‑3)
First: x·x = x²
Outer: x·(‑3) = -3x
Inner: 2·x = 2x
Last: 2·(‑3) = -6
Combine the middle terms: -3x + 2x = -x
Result: x² – x – 6
Special products you should know
These are shortcuts that save time during the exam.
- Square of a binomial: (a+b)² = a² + 2ab + b². Think of it as a square expanding outward.
- Difference of squares: a² – b² = (a+b)(a‑b). It’s like two opposite forces cancelling each other.
Factorisation simplified
Factorising is like taking apart a LEGO model to see which bricks fit together.
Step‑by‑step factor guide
Follow the flowchart above. It keeps you from missing any hidden factor.
Common factor method
Look for a number or variable that appears in every term. Pull it out, then work with the smaller expression.
Example: 6x³ + 9x²
Both terms share 3x². Factor it out: 3x²(2x + 3). Done.
Factorising quadratic trinomials
A quadratic trinomial has the form ax² + bx + c. When a = 1, you just need two numbers that multiply to c and add to b.
Example: x² + 7x + 12
Numbers that multiply to 12 and add to 7 are 3 and 4.
So, x² + 7x + 12 = (x+3)(x+4).
If a ≠ 1, use the “ac method”: multiply a and c, find two numbers that fit, then split the middle term.
Example: 2x² + 5x + 3
ac = 2·3 = 6. Numbers that multiply to 6 and add to 5 are 2 and 3.
Rewrite: 2x² + 2x + 3x + 3 → group: (2x²+2x)+(3x+3) → factor each: 2x(x+1)+3(x+1) → common binomial (x+1): (2x+3)(x+1).
Factorising using special patterns
Remember the difference of squares and perfect square trinomial patterns from the expansion section. They work both ways.
Example: x² – 16 = (x+4)(x‑4) because 16 is 4².
Quick comparison table
| Task | When to use | Key formula |
|---|---|---|
| Expand (a+b)² | Square of a binomial | a² + 2ab + b² |
| Factorise x² – y² | Difference of squares | (x+y)(x‑y) |
| Factorise ax²+bx+c (a=1) | Simple quadratic | (x+m)(x+n) where m·n=c, m+n=b |
| Factorise ax²+bx+c (a≠1) | Complex quadratic | Use ac method |
Tips to avoid common mistakes
- Never forget the sign when you bring a term across the equals sign.
- Check your work by expanding the factorised form – you should get the original polynomial.
- Watch out for a hidden common factor like 2 or x before jumping to special formulas.
📝 Likely Exam Questions
- Expand (3x‑4)(x+5).
Answer: 3x·x + 3x·5 – 4·x – 4·5 = 3x² + 15x – 4x – 20 = 3x² + 11x – 20. - Factorise 2x² – 7x + 3.
Answer: ac = 6, numbers –6 and –1 give –7. Split: 2x² – 6x – x + 3 → 2x(x‑3) –1(x‑3) → (2x‑1)(x‑3). - Simplify (x² – 9) ÷ (x‑3).
Answer: Recognise x² – 9 = (x+3)(x‑3). Cancel (x‑3) → x+3. - Write the factorised form of x² + 6x + 9.
Answer: It’s a perfect square trinomial: (x+3)². - Expand (x‑2)³ (cube of a binomial).
Answer: (x‑2)³ = (x‑2)(x‑2)(x‑2) = x³ – 6x² + 12x – 8.