Expanding And Factorising Brackets

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Lesson notes

Expanding Single Brackets

Expanding means multiplying the term outside the bracket by each term inside.
For example, 3x(x + 2) = 3x × x + 3x × 2 = 3x² + 6x.
Watch out for negative signs: − × − = +, − × + = −.
Use brackets around negative terms to avoid mistakes.

Simplifying After Expanding Single Brackets

First expand each bracket separately, then collect like terms.
Example: 4(x + 7) + 5x(3 − x) = 4x + 28 + 15x − 5x² = 19x + 28 − 5x².
Be careful with subtraction: 3x(x + 2) − 7(x − 6) = 3x² + 6x − 7x + 42 = 3x² − x + 42.

Expanding Double Brackets

Multiply every term in the first bracket by every term in the second.
Use FOIL: First, Outer, Inner, Last.
Example: (x + 1)(x + 3) = x·x + x·3 + 1·x + 1·3 = x² + 4x + 3.
A grid can help organise the multiplication visually.

Grid method for double brackets

Expanding a Bracket Squared

Write (x + 3)² as (x + 3)(x + 3) and expand using FOIL.
Example: (x + 3)² = x² + 3x + 3x + 9 = x² + 6x + 9.
Common mistake: (x + y)² is not x² + y².

Factorising Out Terms

Factorising is the reverse of expanding: write as a product of factors.
Find the highest common factor (HCF) of the number parts and algebra parts.
Place the HCF outside brackets and the remaining terms inside.
Example: 12x² + 18x = 6x(2x + 3).

Factorising with Multiple Variables

Find the HCF of numbers and each variable separately.
Example: 2a³b² − 4a²b³: number HCF = 2, a HCF = a², b HCF = b² → overall HCF = 2a²b².
Result: 2a²b²(a − 2b).
Always check by expanding: factorise fully (e.g., 2x(3x + 5) not x(6x + 10)).

Exam Tips

Check factorisation by expanding the brackets.
Factorise fully means take out the greatest common factor.
Practice with past paper questions: easy, medium, hard.

Slides

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Practice questions

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1.Expand 5(x - 3).
Easy
- A5x - 15
- B5x - 3
- C5x + 15
- D5x + 3
2.Factorise 3x + 12.
Easy
- A3(x + 4)
- B3(x + 12)
- Cx(3 + 12)
- D3x(1 + 4)
3.Factorise 5y - 6py.
Medium
- Ay(5 - 6p)
- By(5 - 6)
- C5y(1 - 6p)
- Dp(5y - 6)
4.Factorise 12x + 15.
Easy
- A3(4x + 5)
- B3(4x + 15)
- C12(x + 15)
- D5(12x + 3)
5.Factorise 5p + pt.
Easy
- Ap(5 + t)
- B5(p + t)
- Cp(5 + pt)
- Dt(5p + 1)
6.Multiply out 9(3 - x).
Easy
- A27 - 9x
- B27 + 9x
- C9x - 27
- D27 - x
7.Factorise completely 18x - 24.
Medium
- A6(3x - 4)
- B3(6x - 8)
- C2(9x - 12)
- D18(x - 6)
8.Expand x(2y + x).
Easy
- A2xy + x²
- B2xy + x
- Cx² + 2y
- D2x²y

Revision notes

## Expanding & Factorising Brackets ### Expanding Single Brackets - Multiply the term outside by **each term** inside. - Watch signs: \(- \times - = +\), \(- \times + = -\). - Example: \(4x(2x-3) = 8x^2 - 12x\). ### Expanding & Simplifying - Expand each bracket separately, then **collect like terms**. - Example: \(2(x+5) + 3x(x-8) = 2x+10+3x^2-24x = 3x^2 -22x +10\). ### Expanding Double Brackets - Use **FOIL** (First, Outer, Inner, Last) or a grid. - \((x+1)(x+3) = x^2 + 4x + 3\). - For squares: \((x+3)^2 = (x+3)(x+3) = x^2+6x+9\). - **Never** write \((x+y)^2 = x^2+y^2\). ### Factorising - Reverse of expanding: put common factor outside brackets. - Find **HCF** of numbers and variables. - Example: \(12x^2+18x = 6x(2x+3)\). - **Factorise fully** means take out the greatest common factor. - Check by expanding. ### Key Points - Always simplify by collecting like terms after expanding. - In factorising, ensure no common factor remains inside brackets.

Expand: Multiply each term inside a bracket by the term outside, removing the brackets.
Factorise: Write an expression as a product of its factors, the reverse of expanding brackets.
Highest Common Factor (HCF): The largest factor that divides all terms; used when factorising.
FOIL: A method for expanding double brackets: multiply First, Outer, Inner, Last terms.
Like terms: Terms with the same variable(s) and power(s); can be combined by adding/subtracting coefficients.

Flashcards

Expand: 5(x - 3)

5x - 15

Multiply out: 9(3 - x)

27 - 9x

Factorise: 5p + pt

p(5 + t)

Factorise: 3x + 12

3(x + 4)

Factorise: 12x + 15

3(4x + 5)

Factorise: 5y - 6py

y(5 - 6p)

Factorise completely: 18x - 24

6(3x - 4)

Expand: x(2y + x)

2xy + x²

Expand and simplify: (x + 3)(x + 5)

x² + 8x + 15

Expand: x²(x - 7)

x³ - 7x²

Factorise: y² + y

y(y + 1)

Factorise: 2x² - x

x(2x - 1)

Factorise completely: 8g² - 4g

4g(2g - 1)

Factorise: 6y + 21

3(2y + 7)

Factorise fully: 10a² - 15ab

5a(2a - 3b)

Expand and simplify: 4(2m + 3) - 5(m - 2)

8m + 12 - 5m + 10 = 3m + 22

Factorise completely: 12b + 8b²

4b(3 + 2b)

Factorise completely: 21a² + 28ab

7a(3a + 4b)

Expand and simplify: (x - 5)(x - 7)

x² - 12x + 35

Multiply out: x(3 - 5x)

3x - 5x²

Factorise fully: 5x - 20x²

5x(1 - 4x)

Factorise completely: 3x² - 12xy

3x(x - 4y)

Expand and simplify: (m - 3)(m + 2)

m² - m - 6

Factorise completely: 6d²e - 9e²

3e(2d² - 3e)

Factorise completely: 10x² + 24x

2x(5x + 12)

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