Factorisation – Definition
Factorisation is the process of breaking a number or an algebraic expression into smaller parts called factors, which when multiplied together give the original number or expression.
🔢 Example with Numbers
If we take the number 12:
12 = 2 × 6
12 = 3 × 4
Here, 2, 3, 4, 6 are factors of 12.
Prime factorisation of 12:
12 = 2 × 2 × 3
✏️ Example with Algebraic Expressions
- 6x = 2 × 3 × x
- x² – 9 = (x – 3)(x + 3)
(This is called factorisation using identity: a² – b² = (a – b)(a + b)) - 2x + 4 = 2(x + 2)
(Taking common factor 2)
Method of factorisation
Factorisation can be done using different methods depending on the type of expression.
1️⃣ Common Factor Method (Taking Out Common Factor)
When all terms have a common number or variable, take it out.
Example:
6x + 9
Common factor = 3
6x + 9 = 3(2x + 3)
Example:
8x²y + 4xy
Common factor = 4xy
= 4xy(2x + 1)
2️⃣ Factorisation by Grouping
Used when there are four terms.
Steps:
- Group terms in pairs
- Take common factors from each group
- Take common binomial
Example:
x² + 3x + 2x + 6
= (x² + 3x) + (2x + 6)
= x(x + 3) + 2(x + 3)
= (x + 3)(x + 2)
3️⃣ Using Identities (Algebraic Identities)
Important Identities:
- a² – b² = (a – b)(a + b)
- (a + b)² = a² + 2ab + b²
- (a – b)² = a² – 2ab + b²
- a³ – b³ = (a – b)(a² + ab + b²)
- a³ + b³ = (a + b)(a² – ab + b²)
Example:
x² – 16
= x² – 4²
= (x – 4)(x + 4)
4️⃣ Splitting the Middle Term Method
Used for quadratic expressions of form:
ax² + bx + c
Steps:
- Multiply a × c
- Find two numbers whose product = ac and sum = b
- Split the middle term
- Factor by grouping
Example:
x² + 5x + 6
Product = 6
Numbers = 2 and 3
= x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
5️⃣ Factorisation of Quadratic When a ≠ 1
Example:
2x² + 7x + 3
Multiply 2 × 3 = 6
Numbers = 6 and 1
= 2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)
✨ Summary Table
| Method | Used When |
|---|---|
| Common Factor | Terms have common number/variable |
| Grouping | Four terms |
| Identities | Expression matches formula |
| Split Middle Term | Quadratic expression |
✅ 20 MCQs on Factorisation
Choose the correct option:
1. The factorisation of 6x + 12 is:
A) 2(3x + 6)
B) 3(2x + 4)
C) 6(x + 2)
D) 12(x + 1)
2. x² – 25 =
A) (x – 5)²
B) (x – 5)(x + 5)
C) (x + 5)²
D) (x – 25)(x + 1)
3. x² + 7x + 10 =
A) (x + 2)(x + 5)
B) (x + 10)(x – 1)
C) (x + 7)(x + 3)
D) (x + 5)(x – 2)
4. 9x² – 16 =
A) (3x – 4)(3x + 4)
B) (9x – 4)(x + 4)
C) (3x – 16)(3x + 1)
D) (9x – 16)(x + 1)
5. 2x² + 9x + 4 =
A) (2x + 1)(x + 4)
B) (2x + 4)(x + 1)
C) (2x + 9)(x + 4)
D) (x + 2)(x + 7)
6. x² – 9x + 20 =
A) (x – 5)(x – 4)
B) (x – 10)(x + 2)
C) (x – 20)(x + 1)
D) (x – 2)(x – 10)
7. The common factor of 8x²y and 12xy² is:
A) 4xy
B) 2xy
C) 8xy
D) 12xy
8. a² + 2ab + b² =
A) (a + b)²
B) (a – b)²
C) (a² + b²)
D) (a + b)(a – b)
9. x³ – 27 =
A) (x – 3)(x² + 3x + 9)
B) (x + 3)(x² – 3x + 9)
C) (x – 27)(x + 1)
D) (x – 3)³
10. 15x + 20 =
A) 5(3x + 4)
B) 3(5x + 20)
C) 15(x + 20)
D) 20(x + 15)
11. x² – 4x – 12 =
A) (x – 6)(x + 2)
B) (x + 6)(x – 2)
C) (x – 4)(x + 3)
D) (x – 12)(x + 1)
12. 4x² – 12x =
A) 4x(x – 3)
B) 2x(2x – 6)
C) x(4x – 12)
D) All of these
13. x² + 11x + 24 =
A) (x + 6)(x + 4)
B) (x + 8)(x + 3)
C) (x + 12)(x + 2)
D) (x + 24)(x + 1)
14. 16a² – b² =
A) (4a – b)(4a + b)
B) (16a – b)(a + b)
C) (4a – b)²
D) (16a – b²)
15. 3x² – 12 =
A) 3(x² – 4)
B) 3(x – 2)(x + 2)
C) Both A and B
D) None
16. x² + 2x – 15 =
A) (x + 5)(x – 3)
B) (x – 5)(x + 3)
C) (x + 15)(x – 1)
D) (x – 15)(x + 1)
17. 5x² + 5x =
A) 5x(x + 1)
B) 5(x² + x)
C) x(5x + 5)
D) All of these
18. x² – 49 =
A) (x – 7)(x + 7)
B) (x – 49)(x + 1)
C) (x – 7)²
D) (x + 49)(x – 1)
19. 6x² + x – 2 =
A) (3x – 2)(2x + 1)
B) (3x + 2)(2x – 1)
C) (6x – 1)(x + 2)
D) (2x – 2)(3x + 1)
20. a³ + 8 =
A) (a + 2)(a² – 2a + 4)
B) (a – 2)(a² + 2a + 4)
C) (a + 8)(a – 1)
D) (a + 2)³
✅ Answer Key
- B
- B
- A
- A
- A
- A
- A
- A
- A
- A
- A
- D
- A
- A
- C
- A
- D
- A
- A
- A
✏️ Factorisation – 2 & 3 Marks Questions
✅ Section A – 2 Marks Questions
(Each question carries 2 marks)
- Factorise:
(a) 8x + 12
(b) 15y – 20 - Factorise using identity:
x² – 36 - Factorise:
a² – 49 - Factorise by taking common factor:
6x² – 9x - Factorise:
x² + 9x + 14 - Factorise:
4a² – 25 - Factorise:
x² – 11x + 28 - Find the common factor of:
12x²y and 18xy²
✅ Section B – 3 Marks Questions
(Each question carries 3 marks)
- Factorise by splitting the middle term:
2x² + 11x + 5 - Factorise:
3x² – 5x – 2 - Factorise using identities:
(a) 9a² – 16b²
(b) x³ – 64 - Factorise by grouping:
x³ + 3x² + 2x + 6 - Factorise completely:
6x² + 7x – 3 - The area of a rectangle is given by
x² + 5x + 6.
Find its length and breadth by factorisation. - If a² + 7a + 12 = 0, factorise the expression and find the values of a.
📘 Case Study Based Questions – Factorisation
🟢 Case Study 1: School Garden Area
A school garden is rectangular. Its area is given by
x² + 7x + 12 square metres.
The school wants to find the length and breadth of the garden by factorisation.
Questions:
- Factorise the expression x² + 7x + 12. (2 marks)
- Find the length and breadth of the garden. (2 marks)
- If x = 3 m, find the actual area of the garden. (1 mark)
🟢 Case Study 2: Playground Design
The area of a playground is given by
9a² – 25 square metres.
Questions:
- Identify the algebraic identity used. (1 mark)
- Factorise 9a² – 25. (2 marks)
- If a = 4 m, find the area of the playground. (2 marks)
🟢 Case Study 3: Profit of a Shopkeeper
The profit earned by a shopkeeper is given by
2x² + 9x + 4.
Questions:
- Factorise the expression by splitting the middle term. (3 marks)
- If x = 2, find the profit. (2 marks)
🟢 Case Study 4: Arrangement of Students
Students are arranged in rows and columns.
Total number of students is given by
x² – 16.
Questions:
- Factorise the expression using identity. (2 marks)
- If x = 6, find the total number of students. (2 marks)
- What type of identity is used? (1 mark)
✅ Answer Key
Case Study 1:
x² + 7x + 12 = (x + 3)(x + 4)
Area when x = 3 → 9 + 21 + 12 = 42 m²
Case Study 2:
Identity: a² – b²
9a² – 25 = (3a – 5)(3a + 5)
When a = 4 → 144 – 25 = 119
Case Study 3:
2x² + 9x + 4 = (2x + 1)(x + 4)
When x = 2 → 8 + 18 + 4 = 30
Case Study 4:
x² – 16 = (x – 4)(x + 4)
When x = 6 → 36 – 16 = 20
Identity: Difference of squares
📝 Mathematics
Chapter: Factorisation
Time: 2 Hours
Maximum Marks: 40
🔹 General Instructions:
- All questions are compulsory.
- Show proper steps for full marks.
- Use of calculator is not permitted.
✅ Section A – MCQs (1 × 8 = 8 Marks)
- x² – 49 =
a) (x – 7)²
b) (x – 7)(x + 7)
c) (x + 7)²
d) (x – 49)(x + 1) - 6x + 9 =
a) 3(2x + 3)
b) 6(x + 9)
c) 9(6x + 1)
d) 3(x + 2) - x² + 5x + 6 =
a) (x + 2)(x + 3)
b) (x – 2)(x – 3)
c) (x + 6)(x – 1)
d) (x + 1)(x + 6) - 4a² – 25 =
a) (2a – 5)(2a + 5)
b) (4a – 5)(a + 5)
c) (2a – 25)(2a + 1)
d) (4a – 25)(a + 1) - 2x² + 7x + 3 =
a) (2x + 1)(x + 3)
b) (2x + 3)(x + 1)
c) (x + 7)(2x + 3)
d) (x + 2)(x + 5) - a² + 2ab + b² =
a) (a – b)²
b) (a + b)²
c) (a + b)(a – b)
d) a² + b² - 9x² – 16 =
a) (3x – 4)(3x + 4)
b) (9x – 4)(x + 4)
c) (3x – 16)(3x + 1)
d) (9x – 16)(x + 1) - Common factor of 12x²y and 18xy² is:
a) 6xy
b) 12xy
c) 3xy
d) 18xy
✅ Section B – Very Short Answer (2 × 6 = 12 Marks)
- Factorise: 8x + 12
- Factorise: x² – 36
- Factorise: x² + 9x + 20
- Factorise: 6x² – 9x
- Factorise: 16a² – b²
- Find the common factor of:
15x²y and 20xy²
✅ Section C – Short Answer (3 × 4 = 12 Marks)
- Factorise by splitting the middle term:
2x² + 11x + 5 - Factorise:
3x² – 5x – 2 - Factorise by grouping:
x³ + 2x² + 3x + 6 - Factorise completely:
6x² + 7x – 3
✅ Section D – Case Study (4 × 2 = 8 Marks)
Case Study:
The area of a rectangular park is given by:
x² + 5x + 6 square metres
- Factorise the expression.
- Find the length and breadth of the park.
- If x = 4 m, find the area.
- Identify the method used in this case.
✅ Answer Key
Section A:
- b
- a
- a
- a
- b
- b
- a
- a
Section B:
9) 4(2x + 3)
10) (x – 6)(x + 6)
11) (x + 4)(x + 5)
12) 3x(2x – 3)
13) (4a – b)(4a + b)
14) 5xy
Section C:
15) (2x + 1)(x + 5)
16) (3x + 1)(x – 2)
17) (x + 2)(x² + 3)
18) (3x – 1)(2x + 3)
Section D:
19) (x + 2)(x + 3)
20) Length = (x + 3), Breadth = (x + 2)
21) 42
22) Splitting the middle term
Leave a Reply