Year 9: Expansion of algebraic expressions

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1. What Is Expansion?

Expansion in algebra means removing brackets by multiplying. It is the reverse of factorisation.

Example: Expand 3(x + 4) ⇒ 3x + 12

2. The Distributive law

The distributive law allows you to multiply a single term by each term inside the brackets:

a(b + c) = ab + ac

Example: 5(x − 2) = 5x − 10

3. Expanding with negative signs

Be careful with signs when expanding:

Example: −2(x − 3) = −2x + 6

Exercises

4. Expanding Two binomials

Use the FOIL method (First, Outer, Inner, Last) to expand (a + b)(c + d):

(x + 2)(x + 5)

• First: x × x = x²
• Outer: x × 5 = 5x
• Inner: 2 × x = 2x
• Last: 2 × 5 = 10

Add all terms: x² + 5x + 2x + 10 = x² + 7x + 10

Exercises

5. Special Products

1. Perfect square: (a ± b)² = a² ± 2ab + b²

Example: (x + 3)² = x² + 6x + 9

2. Difference of squares: (a − b)(a + b) = a² − b²

Example: (x − 4)(x + 4) = x² − 16

6. Practice questions

1. Expand: 4(x + 7)
2. Expand: −3(x − 5)
3. Expand: (x + 2)(x + 6)
4. Expand: (x − 1)(x + 1)
5. Expand: (x − 5)²

Solutions:

1. 4x + 28
2. −3x + 15
3. x² + 6x + 2x + 12 = x² + 8x + 12
4. x² − 1
5. x² − 10x + 25

Practice!

Here are some exercise quizzes to further sharpen your skills in expanding and simplifying:

• Level 1: Simplification and expansion of algebraic expressions
• Level 2: Simplification and expansion of algebraic expressions
• Level 3: Simplification and expansion of algebraic expressions
• Level 1: Simplification of rational algebraic expressions
• Level 2: Simplification of rational algebraic expressions