Expanding And Factorising Brackets

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Notes de leçon

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.

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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

Plus de questions

9.Expand and simplify (x + 3)(x + 5).
Medium
- Ax² + 8x + 15
- Bx² + 15x + 8
- Cx² + 8x + 8
- Dx² + 15
10.Factorise y² + y.
Easy
- Ay(y + 1)
- By(y + 0)
- Cy²(1 + y)
- Dy(1 + y²)
11.Factorise 2x² - x.
Easy
- Ax(2x - 1)
- B2x(x - 1)
- Cx(2x - x)
- D2(x² - x)
12.Factorise completely 8g² - 4g.
Medium
- A4g(2g - 1)
- B2g(4g - 2)
- C4g(2g - 4)
- D8g(g - 0.5)
13.Factorise 6y + 21.
Easy
- A3(2y + 7)
- B3(2y + 21)
- C6(y + 15)
- D21(6y + 1)
14.Factorise fully 10a² - 15ab.
Medium
- A5a(2a - 3b)
- B5a(2a - 3)
- C5(2a² - 3ab)
- D10a(a - 1.5b)
15.Expand the brackets and simplify: 4(2m + 3) - 5(m - 2).
Medium
- A3m + 22
- B3m + 2
- C13m + 22
- D3m - 2
16.Factorise completely 12b + 8b².
Medium
- A4b(3 + 2b)
- B4b(3 + 8b)
- C2b(6 + 4b)
- D8b(1.5 + b)
17.Expand 4x(2x - 3).
Easy
- A8x² - 12x
- B8x² - 3
- C4x² - 12x
- D8x² + 12x
18.Expand -7x(4 - 5y).
Medium
- A-28x + 35xy
- B-28x - 35xy
- C28x + 35xy
- D-28x + 35x
19.Expand and simplify: 2(x + 5) + 3x(x - 8).
Medium
- A3x² - 22x + 10
- B3x² - 24x + 10
- C3x² - 22x + 5
- D2x + 10 + 3x² - 24x
20.Expand and simplify: 3x(x + 2) - 7(x - 6).
Medium
- A3x² - x + 42
- B3x² + 13x + 42
- C3x² + 6x - 7x + 42
- D3x² - x - 42
21.Expand (x + 1)(x + 3).
Medium
- Ax² + 4x + 3
- Bx² + 4x + 4
- Cx² + 3x + 3
- Dx² + 4x + 1
22.Expand (2x - 3)(x + 4).
Medium
- A2x² + 5x - 12
- B2x² + 8x - 12
- C2x² + 5x + 12
- D2x² - 5x - 12
23.Expand (x - 3)(3x - 5).
Medium
- A3x² - 14x + 15
- B3x² - 14x - 15
- C3x² - 8x + 15
- D3x² - 14x + 5
24.Expand (2x + 3)².
Medium
- A4x² + 12x + 9
- B4x² + 9
- C2x² + 6x + 9
- D4x² + 12x + 6
25.Factorise 5x + 15.
Easy
- A5(x + 3)
- B5(x + 15)
- Cx(5 + 15)
- D5x(1 + 3)
26.Factorise fully 30x² - 24x.
Medium
- A6x(5x - 4)
- B6(5x² - 4x)
- Cx(30x - 24)
- D6x(5x - 24)
27.Factorise fully 5x⁴ y³ + 25x³ y³.
Medium
- A5x³ y³(x + 5y)
- B5x⁴ y³(1 + 5x^-1)
- C5x³ y³(x + 5)
- D5x³ y³(x + 5y²)
28.Factorise completely 21a² + 28ab.
Medium
- A7a(3a + 4b)
- B7(3a² + 4ab)
- Ca(21a + 28b)
- D7a(3a + 28b)
29.Expand and simplify (x - 5)(x - 7).
Medium
- Ax² - 12x + 35
- Bx² - 12x - 35
- Cx² + 12x + 35
- Dx² - 2x + 35
30.Expand x(3 - 5x).
Easy
- A3x - 5x²
- B3x - 5x
- C3x + 5x²
- D3 - 5x
31.Factorise fully 5x - 20x².
Medium
- A5x(1 - 4x)
- B5( x - 4x²)
- Cx(5 - 20x)
- D5x(1 - 20x)
32.Factorise completely 3x² - 12xy.
Medium
- A3x(x - 4y)
- B3(x² - 4xy)
- Cx(3x - 12y)
- D3x(x - 12y)
33.The area of a rectangle is given by the expression 12x + 18. If the width of the rectangle is 6, which expression represents the length?
Medium
- A2x + 3
- B2x + 18
- C6x + 3
- D12x + 3
34.A square has side length (x + 2) cm. What is the area of the square in expanded form?
Medium
- Ax² + 4x + 4
- Bx² + 4
- Cx² + 2x + 4
- Dx² + 4x + 2
35.The length of a rectangle is (x + 5) cm and the width is (x - 2) cm. Which expression represents the area of the rectangle in expanded form?
Medium
- Ax² + 3x - 10
- Bx² + 7x - 10
- Cx² + 3x + 10
- Dx² - 3x - 10
36.Expand and simplify: 3(2x + 1) + 4(x - 3).
Medium
- A10x - 9
- B10x + 15
- C6x - 9
- D10x - 12
37.Expand and simplify: (x + 4)(x - 7).
Medium
- Ax² - 3x - 28
- Bx² + 3x - 28
- Cx² - 11x - 28
- Dx² - 3x + 28
38.Expand and simplify: (3x - 2)(x + 5).
Medium
- A3x² + 13x - 10
- B3x² + 17x - 10
- C3x² + 13x + 10
- D3x² - 13x - 10
39.Factorise completely: 8x² - 12x.
Medium
- A4x(2x - 3)
- B2x(4x - 6)
- C4(2x² - 3x)
- D8x(x - 1.5)
40.Factorise completely: 6a² b - 9ab².
Medium
- A3ab(2a - 3b)
- B3a(2ab - 3b²)
- C3b(2a² - 3ab)
- Dab(6a - 9b)
41.Expand 3(2x + 5).
Easy
- A6x + 15
- B6x + 5
- C3x + 15
- D6x + 8
42.Expand and simplify 2(3x - 4) + 5(x + 2).
Medium
- A11x + 2
- B11x - 2
- C6x + 10
- D11x + 18
43.Expanding (x + 3)(x + 4) gives x² + 7x + 12.
Easy
True or false?
44.Expand (x - 2)(x + 5).
Medium
- Ax² + 3x - 10
- Bx² - 3x - 10
- Cx² + 7x - 10
- Dx² - 7x - 10
45.Factorise 8x + 12.
Easy
- A4(2x + 3)
- B2(4x + 6)
- C8(x + 4)
- D4(2x + 12)
46.Factorise completely 15x² - 10x.
Medium
- A5x(3x - 2)
- B5(3x² - 2x)
- Cx(15x - 10)
- D5x(3x + 2)
47.Factorising 4x² - 8x gives 4x(x - 2).
Easy
True or false?
48.Put the steps for expanding (x + 2)(x + 5) in the correct order.
Medium
- Collect like terms: x² + 7x + 10
- Multiply First: x × x = x²
- Multiply Outer: x × 5 = 5x
- Multiply Inner: 2 × x = 2x
- Multiply Last: 2 × 5 = 10
49.Match each expression to its expanded form.
Medium
- 3(2x + 5)
- 4x(3x - 2)
- -2(5 - x)
- 6x + 15
- 12x² - 8x
- -10 + 2x
50.Expanding -3(2x - 5) gives -6x + 15.
Medium
True or false?
51.Expand 5(2x - 3). What is the coefficient of x in the expanded expression?
Easy
52.Expand (3x - 1)(x + 4) and simplify.
Medium
The expanded expression is ____.
53.Factorise 6x² + 9x. What is the numerical coefficient outside the bracket?
Easy
54.Factorise completely 12a² b - 18ab².
Medium
The factorised expression is ____.
55.Complete the expansion.
Easy
Expand 5(2x + 3): ____
56.Expand and simplify (x + 2)(x + 7). What is the coefficient of x in the simplified expression?
Medium
57.Expand 5(2x - 3). What is the coefficient of x in the expanded expression?
Easy
- A12
- B11
- C10
- D9
58.Factorise 6x² + 9x. What is the numerical coefficient outside the bracket?
Easy
- A2
- B3
- C4
- D5
59.Expand and simplify (x + 2)(x + 7). What is the coefficient of x in the simplified expression?
Medium
- A9
- B10
- C8
- D11

Notes de révision

## 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.

Cartes mémo

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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