Expanding Triple Brackets

GCSE Mathematics (Higher)

📦 What are Triple Brackets?

Expanding triple brackets means multiplying three binomial expressions together. The process involves expanding two brackets first, then multiplying the result by the third bracket. This topic is typically for Higher GCSE students.

Example: (x + 1)(x + 2)(x + 3)
Step 1: (x + 1)(x + 2) = x² + 3x + 2
Step 2: (x² + 3x + 2)(x + 3) = x³ + 6x² + 11x + 6

🔢 The Method: Step by Step

Step 1: Expand any two brackets first

Choose any two brackets to expand first using FOIL method.

(x + 1) × (x + 2) = x² + 3x + 2

FOIL expansion:

F: x × x = x²

O: x × 2 = 2x

I: 1 × x = 1x

L: 1 × 2 = 2

x² + 2x + 1x + 2 = x² + 3x + 2

Step 2: Multiply the result by the third bracket

Now multiply your quadratic by the remaining bracket.

(x² + 3x + 2) × (x + 3) = x³ + 6x² + 11x + 6

Multiply each term:

x² × (x + 3) = x³ + 3x²

3x × (x + 3) = 3x² + 9x

2 × (x + 3) = 2x + 6

Add like terms:

x³ + (3x² + 3x²) + (9x + 2x) + 6

= x³ + 6x² + 11x + 6

📚 Types of Triple Brackets

➕ All Brackets Positive

All terms are positive - the most straightforward type.

Example 1: (x + 1)(x + 2)(x + 3)

Step 1: (x + 1)(x + 2) = x² + 3x + 2

Step 2: (x² + 3x + 2)(x + 3)

= x³ + 3x² + 3x² + 9x + 2x + 6

= x³ + 6x² + 11x + 6

Example 2: (x + 2)(x + 3)(x + 4)

Step 1: (x + 2)(x + 3) = x² + 5x + 6

Step 2: (x² + 5x + 6)(x + 4)

= x³ + 4x² + 5x² + 20x + 6x + 24

= x³ + 9x² + 26x + 24

Example 3: (x + 1)(x + 3)(x + 5)

Step 1: (x + 1)(x + 3) = x² + 4x + 3

Step 2: (x² + 4x + 3)(x + 5)

= x³ + 5x² + 4x² + 20x + 3x + 15

= x³ + 9x² + 23x + 15

➖ With Negative Signs

Be careful with signs! Negative × Negative = Positive.

Example 1: (x - 1)(x + 2)(x + 3)

Step 1: (x - 1)(x + 2) = x² + 2x - x - 2 = x² + x - 2

Step 2: (x² + x - 2)(x + 3)

= x³ + 3x² + x² + 3x - 2x - 6

= x³ + 4x² + x - 6

Example 2: (x - 2)(x + 3)(x - 4)

Step 1: (x - 2)(x + 3) = x² + 3x - 2x - 6 = x² + x - 6

Step 2: (x² + x - 6)(x - 4)

= x³ - 4x² + x² - 4x - 6x + 24

= x³ - 3x² - 10x + 24

Example 3: (x - 1)(x - 2)(x - 3)

Step 1: (x - 1)(x - 2) = x² - 2x - x + 2 = x² - 3x + 2

Step 2: (x² - 3x + 2)(x - 3)

= x³ - 3x² - 3x² + 9x + 2x - 6

= x³ - 6x² + 11x - 6

🔢 With Coefficients

Multiply coefficients carefully and use index laws.

Example 1: (2x + 1)(x + 2)(x + 3)

Step 1: (2x + 1)(x + 2) = 2x² + 4x + x + 2 = 2x² + 5x + 2

Step 2: (2x² + 5x + 2)(x + 3)

= 2x³ + 6x² + 5x² + 15x + 2x + 6

= 2x³ + 11x² + 17x + 6

Example 2: (3x - 1)(x + 2)(2x - 3)

Step 1: (3x - 1)(x + 2) = 3x² + 6x - x - 2 = 3x² + 5x - 2

Step 2: (3x² + 5x - 2)(2x - 3)

= 6x³ - 9x² + 10x² - 15x - 4x + 6

= 6x³ + x² - 19x + 6

⬆️ Perfect Cubes

When all three brackets are the same: (x + a)³

Example 1: (x + 2)³

(x + 2)³ = (x + 2)(x + 2)(x + 2)

Step 1: (x + 2)(x + 2) = x² + 4x + 4

Step 2: (x² + 4x + 4)(x + 2)

= x³ + 2x² + 4x² + 8x + 4x + 8

= x³ + 6x² + 12x + 8

Formula: (x + a)³ = x³ + 3ax² + 3a²x + a³

Example 2: (x - 1)³

(x - 1)³ = (x - 1)(x - 1)(x - 1)

Step 1: (x - 1)(x - 1) = x² - 2x + 1

Step 2: (x² - 2x + 1)(x - 1)

= x³ - x² - 2x² + 2x + x - 1

= x³ - 3x² + 3x - 1

🧮 Interactive Triple Bracket Expander

Enter your own triple brackets and see them expand step-by-step!

Bracket 1:

Bracket 2:

Bracket 3:

✏️ Practice Triple Brackets

Expand: (x + 1)(x + 2)(x + 3)

Correct: 0 Attempted: 0

🎯 Common Patterns to Recognize

Pattern 1: Consecutive numbers

(x + n)(x + n+1)(x + n+2)

(x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6

The coefficients follow a pattern: 1, 6, 11, 6

Pattern 2: Symmetric brackets

(x + a)(x + b)(x + c) where a + b + c = constant

(x + 1)(x + 2)(x + 4) = x³ + 7x² + 14x + 8

Notice: 1 + 2 + 4 = 7 (coefficient of x²)

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