Factorisation is an important topic in algebra that students learn in Classes 7 to 10. It is the method of breaking down a number or a polynomial into smaller parts called factors. These factors, when multiplied together, give back the original number or expression. For example, the factorisation of the expression x² + 2x is x(x + 2). Here, x and (x + 2) are the factors.

Learning how to factorise helps in solving equations easily and is very useful in higher mathematics. In this topic, you will learn how to find common terms, use identities, and rearrange terms to break down complex expressions. In this article, we’ll go through some solved factorisation problems to help you understand the steps clearly. You’ll also find a few practice questions at the end so you can test how well you’ve understood the concept.

Q.1: Factorise 6x² + 19x + 10

Solution:
We break the middle term 19x into two parts:
→ 6x² + 15x + 4x + 10

Now, take out common terms from each pair:
→ 3x(2x + 5) + 2(2x + 5)

Take out the common factor:
→ (3x + 2)(2x + 5)

Q.2: Factorise z² + 13z + 30

Solution:
Split the middle term 13z into 10z and 3z:
→ z² + 10z + 3z + 30

Group and factor:
→ z(z + 10) + 3(z + 10)

Final answer:
→ (z + 3)(z + 10)

Q.3: Factorise 3x² + 10x – 8

Solution:
Split the middle term 10x into 12x and -2x:
→ 3x² + 12x – 2x – 8

Take out the common terms:
→ 3x(x + 4) – 2(x + 4)

Now factor:
→ (3x – 2)(x + 4)

Q.4: Factorise 9(m + n)² – 18(m² – n²) + 16(m – n)²

Solution:
We know that m² – n² = (m + n)(m – n)
Now substitute that in:
→ 9(m + n)² – 18(m + n)(m – n) + 16(m – n)²

Now group as a perfect square:
→ [3(m + n) – 4(m – n)]²
→ [3m + 3n – 4m + 4n]²
→ (7n – m)²

Q.5: Factorise 36(p + q)² – (p – q)²

Solution:
Write the expression using squares:
→ [6(p + q)]² – (p – q)²

Use the identity a² – b² = (a + b)(a – b):
→ [6(p + q) + (p – q)] [6(p + q) – (p – q)]

Simplify both:
→ [6p + 6q + p – q] [6p + 6q – p + q]
→ [7p + 5q][5p + 7q]

Q.6: Factor 8m²n – 12mn + 6mn²

Solution:
Find the HCF, which is 2mn:
→ 2mn(4m – 6 + 3n)

Q.7: Simplify (9x² – 16y²)/(3x + 4y)

Solution:
Recognise it as a difference of squares:
→ (3x)² – (4y)² = (3x + 4y)(3x – 4y)

Now cancel (3x + 4y):
→ [(3x + 4y)(3x – 4y)] / (3x + 4y)

Final answer:
→ 3x – 4y

Additional Factorisation Practice Questions

1. Factor: 20x³ + 5x² – 10x
2. Factorise: 64x² − 40x + 6
3. Find the factors of: 9x²yz − 7y
4. Factor: x² − 13x + 36
5. Factorise: 13x² + 39x − 130
6. Factorise: −25 + 64x²
7. Find the factors of the expression: 98 − 2x²
8. Find the factors of: 16x² + 25y² + 40xy
9. Factorise using algebraic identities: x² + 10x + 25
10. Given that one of the factors of (6x² + 60x − 144) is (x − 3), find the other factor.