Expansion & Factorisation: The Complete O-Level Guide

Expansion and factorisation are two sides of the same coin — and they’re the foundation of almost every algebra topic you’ll meet at O-Level. Master them here with worked examples, special product shortcuts, and an interactive factoriser.

The Big Picture: Expansion vs Factorisation

Think of expansion and factorisation as opposite operations — like multiplication and division.

Operation	What It Does	Direction
Expansion	Removes brackets	$3(x + 2) \rightarrow 3x + 6$
Factorisation	Puts brackets back	$3x + 6 \rightarrow 3(x + 2)$

Every factorisation can be checked by expanding — if you get back the original expression, you’re correct.

💡 The Golden Check

Always verify your factorisation by expanding the answer. If it matches the original expression, you’ve nailed it.

Part 1: Expansion

Expanding Single Brackets (Distributive Law)

The distributive law says: multiply the term outside the bracket by every term inside.

$a(b + c) = ab + ac$

Example 1: Single Bracket Expansion

Problem:

Expand $-4(3x - 5)$

Step 1: Multiply $-4$ by the first term: $-4 \times 3x = -12x$

Step 2: Multiply $-4$ by the second term: $-4 \times (-5) = +20$

$-4(3x - 5) = -12x + 20$

Example 2: Expand and Simplify

Problem:

Expand and simplify $5(2x + 3) - 3(x - 4)$

Step 1: Expand the first bracket: $5(2x + 3) = 10x + 15$

Step 2: Expand the second bracket: $-3(x - 4) = -3x + 12$

Step 3: Combine like terms: $10x + 15 - 3x + 12 = 7x + 27$

$5(2x + 3) - 3(x - 4) = 7x + 27$

⚠️ Sign Trap

When a minus sign sits before a bracket, it flips every sign inside. $-(x - 3) = -x + 3$ , not $-x - 3$ .

Expanding Double Brackets (FOIL Method)

To expand $(a + b)(c + d)$ , multiply each term in the first bracket by each term in the second. The FOIL mnemonic helps:

Letter	Stands For	Multiply
F	First	$a \times c$
O	Outer	$a \times d$
I	Inner	$b \times c$
L	Last	$b \times d$

$(a + b)(c + d) = ac + ad + bc + bd$

Example 3: FOIL in Action

Problem:

Expand $(x + 3)(x + 5)$

Step	Pair	Result
F	$x \times x$	$x^2$
O	$x \times 5$	$5x$
I	$3 \times x$	$3x$
L	$3 \times 5$	$15$

Combine like terms ( $5x + 3x = 8x$ ):

$(x + 3)(x + 5) = x^2 + 8x + 15$

Example 4: FOIL with Negatives

Problem:

Expand $(2x - 3)(x + 7)$

Step	Pair	Result
F	$2x \times x$	$2x^2$
O	$2x \times 7$	$14x$
I	$-3 \times x$	$-3x$
L	$-3 \times 7$	$-21$

Combine like terms ( $14x - 3x = 11x$ ):

$(2x - 3)(x + 7) = 2x^2 + 11x - 21$

Three Special Products You Must Memorise

These identities appear so often in O-Level that you should recognise them instantly — both directions (expansion and factorisation).

The Three Special Products

1. Perfect Square (Sum)

$(a + b)^2 = a^2 + 2ab + b^2$

2. Perfect Square (Difference)

$(a - b)^2 = a^2 - 2ab + b^2$

3. Difference of Two Squares

$(a + b)(a - b) = a^2 - b^2$

💡 Why the Middle Term Vanishes

In $(a + b)(a - b)$ , the outer and inner products are $-ab$ and $+ab$ — they cancel out, leaving only $a^2 - b^2$ . That’s why there’s no middle term.

Example 5: Perfect Square Expansion

Problem:

Expand $(3x + 4)^2$

Using $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 3x$ and $b = 4$ :

$a^2 = (3x)^2 = 9x^2$
$2ab = 2(3x)(4) = 24x$
$b^2 = 4^2 = 16$

$(3x + 4)^2 = 9x^2 + 24x + 16$

Example 6: Difference of Two Squares

Problem:

Expand $(5x + 2)(5x - 2)$

This matches the pattern $(a + b)(a - b) = a^2 - b^2$ with $a = 5x$ and $b = 2$ :

$(5x + 2)(5x - 2) = (5x)^2 - 2^2 = 25x^2 - 4$

No middle term — just two squares with a minus between them.

Part 2: Factorisation

Factorisation is the reverse of expansion. You’re putting brackets back into an expression.

Method 1: Extracting Common Factors

Look for a factor shared by every term, then pull it outside.

Example 7: Common Factor

Problem:

Factorise $12x^3 - 8x^2 + 4x$

Step 1: Find the HCF of the coefficients: HCF of 12, 8, 4 = 4

Step 2: Find the lowest power of $x$ : $x^3$ , $x^2$ , $x$ → lowest is $x$

Step 3: Common factor is $4x$ . Divide each term:

$12x^3 \div 4x = 3x^2$
$-8x^2 \div 4x = -2x$
$4x \div 4x = 1$

$12x^3 - 8x^2 + 4x = 4x(3x^2 - 2x + 1)$

⚠️ Don't Forget the 1

When you factor out the entire last term, a 1 must remain inside the bracket. $6x + 6 = 6(x + 1)$ , not $6(x)$ .

Method 2: Factorising Quadratics ( $x^2 + bx + c$ )

When $a = 1$ , find two numbers that multiply to $c$ and add to $b$ .

$x^2 + bx + c = (x + p)(x + q) \quad \text{where } p \times q = c \text{ and } p + q = b$

Example 8: Factorise x² + bx + c (both positive)

Problem:

Factorise $x^2 + 7x + 12$

We need two numbers that multiply to 12 and add to 7.

Factor pair of 12	Sum
$1 \times 12$	$13$
$2 \times 6$	$8$
$3 \times 4$	$7$ ✓

$x^2 + 7x + 12 = (x + 3)(x + 4)$

Example 9: Factorise x² + bx + c (negative constant)

Problem:

Factorise $x^2 + 2x - 15$

We need two numbers that multiply to $-15$ and add to $+2$ .

Since the product is negative, one number is positive and the other is negative:

Factor pair of $-15$	Sum
$-1 \times 15$	$14$
$1 \times (-15)$	$-14$
$-3 \times 5$	$2$ ✓

$x^2 + 2x - 15 = (x - 3)(x + 5)$

Sign Shortcut for $x^2 + bx + c$

$c$ is…	$b$ is…	Signs in brackets
Positive (+)	Positive (+)	$(x + \text{\_})(x + \text{\_})$
Positive (+)	Negative (−)	$(x - \text{\_})(x - \text{\_})$
Negative (−)	Positive (+)	$(x + \text{big})(x - \text{small})$
Negative (−)	Negative (−)	$(x - \text{big})(x + \text{small})$

Method 3: Factorising Quadratics ( $ax^2 + bx + c$ where $a \neq 1$ )

When the leading coefficient isn’t 1, use the cross method (also called the AC method):

Find two numbers that multiply to $a \times c$ and add to $b$ .
Rewrite the middle term using those two numbers.
Factorise by grouping.

Example 10: Cross Method (a ≠ 1)

Problem:

Factorise $2x^2 + 7x + 3$

Step 1: Calculate $a \times c = 2 \times 3 = 6$

Find two numbers that multiply to 6 and add to 7: $1$ and $6$

Step 2: Split the middle term: $2x^2 + x + 6x + 3$

Step 3: Factor by grouping:

Group 1: $2x^2 + x = x(2x + 1)$
Group 2: $6x + 3 = 3(2x + 1)$

Step 4: Extract the common bracket:

$2x^2 + 7x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)$

Check: $(x + 3)(2x + 1) = 2x^2 + x + 6x + 3 = 2x^2 + 7x + 3$ ✓

Example 11: Cross Method with Negatives

Problem:

Factorise $3x^2 - 11x + 6$

Step 1: $a \times c = 3 \times 6 = 18$

Find two numbers that multiply to $18$ and add to $-11$ : $-2$ and $-9$

Step 2: Split: $3x^2 - 2x - 9x + 6$

Step 3: Group:

$3x^2 - 2x = x(3x - 2)$
$-9x + 6 = -3(3x - 2)$

Step 4: Extract:

$3x^2 - 11x + 6 = (x - 3)(3x - 2)$

Check: $(x - 3)(3x - 2) = 3x^2 - 2x - 9x + 6 = 3x^2 - 11x + 6$ ✓

Try It Yourself: Interactive Factoriser

Type any quadratic expression below to see the complete factorisation with cross method steps.

Quadratic Factorizer

Enter coefficients to factor ax² + bx + c into (px + r)(qx + s)

Try an example:

x²

Method 4: Difference of Two Squares

Spot the pattern: two perfect squares separated by a minus sign.

$a^2 - b^2 = (a + b)(a - b)$

Example 12: Difference of Two Squares

Problem:

Factorise $49x^2 - 16$

Recognise: $49x^2 = (7x)^2$ and $16 = 4^2$

$49x^2 - 16 = (7x)^2 - 4^2 = (7x + 4)(7x - 4)$

Example 13: Hidden Difference of Squares

Problem:

Factorise $3x^2 - 75$

Step 1: Extract common factor first: $3x^2 - 75 = 3(x^2 - 25)$

Step 2: Now factorise the difference of squares: $x^2 - 25 = (x + 5)(x - 5)$

$3x^2 - 75 = 3(x + 5)(x - 5)$

❌ Sum of Squares Cannot Be Factorised

$a^2 + b^2$ does not factorise over the real numbers. $x^2 + 9 \neq (x + 3)(x - 3)$ . Only the difference works.

Method 5: Perfect Square Trinomials

If you spot $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ , it folds neatly into a squared bracket.

Example 14: Perfect Square Trinomial

Problem:

Factorise $x^2 - 10x + 25$

Check: Is this a perfect square trinomial?

First term: $x^2 = (x)^2$ ✓
Last term: $25 = (5)^2$ ✓
Middle term: $-10x = -2(x)(5)$ ✓

$x^2 - 10x + 25 = (x - 5)^2$

Example 15: Perfect Square with a ≠ 1

Problem:

Factorise $4x^2 + 12x + 9$

Check:

First term: $4x^2 = (2x)^2$ ✓
Last term: $9 = (3)^2$ ✓
Middle term: $12x = 2(2x)(3)$ ✓

$4x^2 + 12x + 9 = (2x + 3)^2$

Choosing the Right Method: A Decision Flowchart

When you see a factorisation question, follow this order:

Factorisation Decision Order

Common factor? Always check this first. Pull out the HCF.

Two terms? Check for difference of two squares: $a^2 - b^2$ .

Three terms? Check for perfect square trinomial, then try the factor-pair / cross method.

Four terms? Try factorisation by grouping (pair the terms).

5 Common Mistakes to Avoid

Mistake 1: Forgetting to Distribute the Negative

Wrong: $-(3x - 2) = -3x - 2$

Right: $-(3x - 2) = -3x + 2$

The minus sign flips both signs inside the bracket.

Mistake 2: Squaring a Bracket Incorrectly

Wrong: $(x + 3)^2 = x^2 + 9$

Right: $(x + 3)^2 = x^2 + 6x + 9$

You can’t just square each term — you’re missing the $2ab$ middle term. Use the identity or FOIL it out.

Mistake 3: Confusing Signs in Factor Pairs

When factorising $x^2 - x - 12$ , you need two numbers that multiply to $-12$ and add to $-1$ .

Wrong: $(x - 3)(x - 4)$ → product is $+12$ , not $-12$

Right: $(x + 3)(x - 4)$ → product is $-12$ , sum is $-1$ ✓

Mistake 4: Stopping at Common Factor

$2x^2 - 18 = 2(x^2 - 9)$ … but you’re not done!

Complete answer: $2x^2 - 18 = 2(x + 3)(x - 3)$

Always check if the expression inside the brackets can be factorised further.

Mistake 5: Factorising a Sum of Squares

$x^2 + 25 \neq (x + 5)(x - 5)$

$(x + 5)(x - 5) = x^2 - 25$ , not $x^2 + 25$ . The sum of two squares cannot be factorised.

Quick Reference Table

Expression Type	Formula	Example
Single bracket	$a(b + c) = ab + ac$	$3(x + 4) = 3x + 12$
Double bracket (FOIL)	$(a+b)(c+d) = ac+ad+bc+bd$	$(x+2)(x+5) = x^2+7x+10$
Perfect square (sum)	$(a+b)^2 = a^2+2ab+b^2$	$(x+3)^2 = x^2+6x+9$
Perfect square (diff)	$(a-b)^2 = a^2-2ab+b^2$	$(x-4)^2 = x^2-8x+16$
Difference of squares	$(a+b)(a-b) = a^2-b^2$	$(x+6)(x-6) = x^2-36$
Common factor	$ab + ac = a(b+c)$	$6x+9 = 3(2x+3)$
Quadratic ( $a=1$ )	$x^2+bx+c = (x+p)(x+q)$	$x^2+5x+6 = (x+2)(x+3)$
Quadratic ( $a \neq 1$ )	Use cross / AC method	$2x^2+5x+3 = (2x+3)(x+1)$

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Expansion & Factorisation: The Complete O-Level Guide

Expansion & Factorisation: The Complete O-Level Guide

The Big Picture: Expansion vs Factorisation

Part 1: Expansion

Expanding Single Brackets (Distributive Law)

Expanding Double Brackets (FOIL Method)

Three Special Products You Must Memorise

The Three Special Products

Part 2: Factorisation

Method 1: Extracting Common Factors

Method 2: Factorising Quadratics ( $x^2 + bx + c$ )

Sign Shortcut for $x^2 + bx + c$

Method 3: Factorising Quadratics ( $ax^2 + bx + c$ where $a \neq 1$ )

Try It Yourself: Interactive Factoriser

Quadratic Factorizer

Method 4: Difference of Two Squares

Method 5: Perfect Square Trinomials

Choosing the Right Method: A Decision Flowchart

Factorisation Decision Order

5 Common Mistakes to Avoid

Mistake 1: Forgetting to Distribute the Negative

Mistake 2: Squaring a Bracket Incorrectly

Mistake 3: Confusing Signs in Factor Pairs

Mistake 4: Stopping at Common Factor

Mistake 5: Factorising a Sum of Squares

Quick Reference Table

Ready to Practice Expansion & Factorisation?

Topics covered:

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Expansion & Factorisation: The Complete O-Level Guide

The Big Picture: Expansion vs Factorisation

Part 1: Expansion

Expanding Single Brackets (Distributive Law)

Expanding Double Brackets (FOIL Method)

Three Special Products You Must Memorise

The Three Special Products

Part 2: Factorisation

Method 1: Extracting Common Factors

Method 2: Factorising Quadratics (x2+bx+cx^2 + bx + cx2+bx+c)

Sign Shortcut for x2+bx+cx^2 + bx + cx2+bx+c

Method 3: Factorising Quadratics (ax2+bx+cax^2 + bx + cax2+bx+c where a≠1a \neq 1a=1)

Try It Yourself: Interactive Factoriser

Quadratic Factorizer

Method 4: Difference of Two Squares

Method 5: Perfect Square Trinomials

Choosing the Right Method: A Decision Flowchart

Factorisation Decision Order

5 Common Mistakes to Avoid

Mistake 1: Forgetting to Distribute the Negative

Mistake 2: Squaring a Bracket Incorrectly

Mistake 3: Confusing Signs in Factor Pairs

Mistake 4: Stopping at Common Factor

Mistake 5: Factorising a Sum of Squares

Quick Reference Table

Ready to Practice Expansion & Factorisation?

Topics covered:

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Method 2: Factorising Quadratics ( $x^2 + bx + c$ )

Sign Shortcut for $x^2 + bx + c$

Method 3: Factorising Quadratics ( $ax^2 + bx + c$ where $a \neq 1$ )