Class 8 Mathematics Chapter 12 Factorisation

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I. Chapter Summary:

This chapter introduces students to the concept of factorisation, which means expressing a number or an algebraic expression as a product of its factors. Students learn to factorise expressions using common factors, identities, grouping methods, and irreducible polynomials. The chapter develops the foundation for higher algebra, polynomial operations, and equation solving in later grades.

II. Key Concepts Covered:

Concept	Explanation
Factors and Multiples	Understanding the difference between factors and multiples in arithmetic and algebra.
Factorisation by Common Factors	Taking out the highest common factor (HCF) from algebraic expressions.
Factorisation by Regrouping	Grouping terms in such a way that a common factor emerges from each group.
Factorisation Using Identities	Using standard identities like:
- $a^2 – b^2 = (a + b)(a – b)$
- $a + b)^2 = a^2 + 2ab + b^2$
Irreducible Polynomials	Expressions that cannot be factorised further using rational numbers.
Division of Algebraic Expressions	Dividing a polynomial by monomial or binomial.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

Factorise: $x^2 – 9$
a) $x + 3)^2$
b) $(x + 9) (x - 9)$
c) $(x + 3) (x - 3)$
d) $(x + 1) (x - 1)$
Answer: c) $(x + 3) (x - 3)$
Which of the following is not factorised form?
a) $3 x (x + 5)$
b) $x^2 + 5x$
c) $(x + 2) (x - 2)$
d) $2 (x - 3)$
Answer: b) $x^2 + 5x$
(PYQ 2020) The factors of $2 x y + 4 x$ are:
a) $2 x (y + 2)$
b) $x (2 y + 4)$
c) $2 (x + y)$
d) None of these
Answer: a) $2 x (y + 2)$
What is the identity used in factorising $x^2 – 16$ ?
a) $a^2 + b^2$
b) $a + b)^2$
c) $a – b)^2$
d) $a^2 – b^2 = (a + b)(a – b)$
Answer: d)

(B) Short Answer Questions (2–3 Marks)

Factorise using common factors: $3x^2y + 6xy^2$
Factorise: $x^2 + 5x + 6$
(PYQ 2019) Factorise the expression: $2×2+6x2x^2 + 6x$
Factorise by regrouping: $ab + a + b + 1$

(C) Long Answer Questions (5 Marks)

Factorise completely: $4×2−25y24x^2 – 25y^2$ and explain each step.
Using identities, factorise: $x^2 + 6x + 9$
(PYQ 2018) Divide: $x^2 + 5x + 6)$ by $(x + 3)$ and state the quotient.
Factorise: $2×2+7x+32x^2 + 7x + 3$ using the middle-term splitting method.

(D) HOTS – Higher Order Thinking Skills

A polynomial $x^2 + ax + b$ is divisible by $(x + 2)$ . Find values of $a$ and $b$ , and hence factorise.
Factorise and verify: $x + y)^2 – 4xy$

IV. Key Formulas / Concepts

Concept	Formula
Common Factor Rule	Take HCF of coefficients and variables.
Algebraic Identities Used for Factorisation
- $a^2 – b^2 = (a + b)(a – b)$
- $a^2 + 2ab + b^2 = (a + b)^2$
- $a^2 – 2ab + b^2 = (a – b)^2$
Middle-Term Splitting	Splitting middle term to factor trinomials.
Grouping Method	Grouping terms to factor in pairs.
Polynomial Division by Monomial	Divide each term individually by the monomial.

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Factorisation	6–8 Marks	Factorise expressions, use identities, HOTS, word problems

VII. Previous Year Questions (PYQs)

Year	Marks	Question
2020	1 mark	Factorise $2 x y + 4 x$
2019	3 marks	Factorise $2×2+6x2x^2 + 6x$
2018	5 marks	Divide polynomial using binomial
2017	2 marks	Factorise using identity

VIII. Real-World Application Examples

Area of Geometrical Shapes: Factorising area expressions to find unknown dimensions.
Problem Solving in Physics and Economics: Simplifying formulas using factorisation.
Coding & Algorithms: Factorisation used in computer algorithms and encryption.
Construction and Design: Rearranging algebraic models for calculations in civil and mechanical engineering.

IX. Student Tips & Strategies for Success (Class 8 Specific)

Time Management:

Allocate 15 minutes daily to practice 3–4 factorisation questions.
Use a weekly checklist to cover all types: identities, common factor, regrouping.

Exam Preparation:

Maintain a “Factor Bank” notebook with formulas and solved examples.
Practice previous year questions and school sample papers regularly.

Stress Management:

Use color-coding (e.g., underline identities, circle HCF) to reduce errors.
Take 5-minute mental math breaks with flashcards or puzzle games.

X. Career Guidance & Exploration

For Classes 9–10:

Mastering factorisation builds foundation for algebra, quadratic equations, and polynomials.
Helpful for Math Olympiads, NTSE, and logical reasoning tests.

For Classes 11–12:

Critical in JEE, CUET, NDA, and NEET exams.
Used extensively in fields such as:
- Computer Science (e.g., algorithms)
- Engineering (structural formulas)
- Economics (profit/loss algebra)
- Data Science (equation modeling)

XI. Important Notes

Don’t skip checking by expanding the final factorised form.
Always look for common terms before applying identities.
Divide polynomial step-by-step; avoid skipping terms in long division.

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