Expanding And Factorising Brackets

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टीचर्स के लिए: Expanding And Factorising Brackets (Maths [CIE], Core) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स, डायग्राम — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

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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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 59 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

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

और सवाल

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

रिवीज़न नोट्स

## 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.

फ्लैशकार्ड

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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