Why expansions and factorisation matter
Ever tried to solve a puzzle where the pieces keep changing shape? That's what algebra feels like until you master expansions and factorisation. They let you flip expressions back and forth, making tough problems simple.
💡 In Simple Words: Expanding is like opening a gift box – you spread out everything inside. Factorising is the reverse – you pack the pieces back into neat groups. Both tricks help you see hidden patterns.
Understanding Expansions
Expansion means rewriting a product of two or more brackets into a single sum of terms. Think of it as pouring water from several small bottles into one big jug – the amount stays the same, just the container changes.
Key formulas you’ll use
- (a + b)(c + d) = ac + ad + bc + bd – spread each part of the first bracket across each part of the second.
- (a + b)² = a² + 2ab + b² – a special case called a perfect square, where the same two numbers are multiplied.
- (a – b)² = a² – 2ab + b² – same idea, but the middle term flips sign.
- (a + b)(a – b) = a² – b² – known as the difference of squares, because the b² term disappears.
Worked example: expand (x + 5)(x – 2)
Step 1: Write each term of the first bracket next to each term of the second.
(x + 5)(x – 2) = x·x + x·(–2) + 5·x + 5·(–2)
Step 2: Multiply.
= x² – 2x + 5x – 10
Step 3: Combine like terms (terms that have the same variable part).
= x² + 3x – 10
So the expanded form is x² + 3x – 10.
Understanding Factorisation
Factorisation is the opposite of expansion – you rewrite a sum of terms as a product of brackets. Imagine you have a pile of LEGO bricks (the sum) and you want to sort them into two distinct colour groups (the brackets) that can be snapped together.
When to factorise
You usually factorise when:
- The expression is part of an equation you need to solve.
- You want to simplify a fraction.
- It matches a known pattern like a perfect square or difference of squares.
Common factor method
First, look for a common factor – a number or variable that appears in every term. Pull it out, just like taking the same size slice from every piece of pizza.
Grouping method
If there’s no single common factor, try to split the expression into two pairs, factor each pair, then look for a common binomial (two-term expression) between the pairs.
Special patterns
- Difference of squares: a² – b² = (a + b)(a – b)
- Perfect square: a² + 2ab + b² = (a + b)²
- Sum of cubes: a³ + b³ = (a + b)(a² – ab + b²) – a bit rarer for Class 9 but good to recognise.
Worked example: factorise 2x² + 7x + 3
Step 1: Look for a common factor. None works for all three terms.
Step 2: Find two numbers that multiply to (coefficient of x² × constant term) = 2 × 3 = 6 and add to the middle coefficient 7.
Those numbers are 6 and 1.
Step 3: Rewrite the middle term using those numbers.
2x² + 6x + x + 3
Step 4: Group and factor each pair.
(2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)
Step 5: Pull out the common binomial (x + 3).
= (x + 3)(2x + 1)
So the factorised form is (x + 3)(2x + 1).
Quick Comparison: Expansion vs Factorisation
| Goal | Typical starting form | Key step | Result |
|---|---|---|---|
| Expand | Product of brackets | Apply distributive law (multiply each term) | Single polynomial sum |
| Factorise | Polynomial sum | Find common factor or pattern, then write as product | Product of brackets |
Factorisation flowchart
📝 Likely Exam Questions
- Expand (3x – 4)(2x + 5).
Answer: 6x² + 7x – 20 - Factorise x² – 9.
Answer: (x + 3)(x – 3) - Write the factorised form of 4y² – 12y + 9.
Answer: (2y – 3)² - Solve 2x² + 7x + 3 = 0 by factorisation.
Answer: (x + 3)(2x + 1) = 0 → x = –3 or x = –½ - Explain why 5a² – 20 can be written as 5(a² – 4).
Answer: 5 is a common factor of both terms; factoring it out simplifies the expression.