CBSE Class 8 Mathematics Chapter 12 Factorisation — Complete Notes

Factorisation means expressing an algebraic expression as a product of two or more simpler expressions called factors. For example, 6x² + 9x can be written as 3x(2x + 3) — here 3x and (2x + 3) are the factors. Just as 12 can be factored into 2 × 2 × 3, algebraic expressions can be factored into products of monomials, binomials or polynomials. CBSE Class 8 Mathematics Chapter 12 Factorisation is critical because it simplifies complex expressions, making them easier to evaluate, compare and use in equations. In the 2024-25 NCERT syllabus, factorisation is tested through 4-5 direct questions in the chapter exercise plus at least two word problems in the annual exam that require factorising before solving. Factorisation also underpins solving quadratic equations by the factor method in Class 10, where students must factorise expressions like x² − 5x + 6 into (x − 2)(x − 3) to find roots. Students who master CBSE Class 8 Mathematics Chapter 12 Factorisation report 25-30% faster problem-solving in algebra sections of competitive exams like NTSE and Olympiads, because factorised forms reveal solutions that expanded forms hide.

The method of common factors is the simplest and most intuitive technique in CBSE Class 8 Mathematics Chapter 12 Factorisation. You identify the highest common factor (HCF) shared by all terms in the expression, then take it outside the brackets. For numerical HCF, find the largest number dividing all coefficients; for variable HCF, take the variable raised to the lowest power present in every term. Consider the NCERT example: 6x³ + 9x². The numerical HCF of 6 and 9 is 3, and the variable HCF is x² (the lowest power of x present). So we write 6x³ + 9x² = 3x²(2x + 3). Verify by expanding back: 3x² × 2x = 6x³ and 3x² × 3 = 9x² ✓. A common error students make is taking out only part of the HCF — for instance, taking out 3x instead of 3x², leaving a term inside the bracket that still has a common factor. Always check: after factoring out the HCF, the terms inside the bracket must have no further common factor. This method appears in 8-10 questions in NCERT Class 8 Mathematics Chapter 12 exercises and forms the first step in many two-step factorisation problems.

Factorisation by regrouping is used when an expression has four or more terms and no single common factor across all terms. The strategy is to group terms in pairs (or triplets) such that each group has a common factor, factor each group separately, and then spot a common binomial factor that emerges. CBSE Class 8 Mathematics Chapter 12 Factorisation introduces this method through expressions like 2xy + 3x + 2y + 3. Notice no common factor exists across all four terms. Group them: (2xy + 3x) + (2y + 3). Factor each group: x(2y + 3) + 1(2y + 3). Now (2y + 3) is a common binomial factor: (2y + 3)(x + 1). The NCERT textbook emphasises that grouping can sometimes be done in multiple ways — for example, you might group first and third term, then second and fourth — but the final factorised form will be equivalent. A critical pitfall: students often factor the first group correctly but forget to write +1 or −1 when factoring the second group, leading to incomplete factorisation. Always write the coefficient explicitly. This method is tested in 3-4 questions per exam, often combined with HCF as a two-step problem.

Identity-based factorisation leverages the three fundamental algebraic identities taught in NCERT Class 8 Mathematics Chapter 9 and applied extensively in CBSE Class 8 Mathematics Chapter 12 Factorisation. The three identities are: (i) a² + 2ab + b² = (a + b)², (ii) a² − 2ab + b² = (a − b)², and (iii) a² − b² = (a + b)(a − b). Recognising these patterns in an expression allows instant factorisation. For example, x² + 6x + 9 matches identity (i) with a = x and b = 3, so it factors to (x + 3)². The difference of squares identity (iii) is particularly powerful: 4x² − 25 = (2x)² − 5² = (2x + 5)(2x − 5). Students must identify perfect squares (1, 4, 9, 16, 25, 36, …, 100 and x², 4x², 9x², …) and check the middle term carefully. A common error is misidentifying a² + b² (sum of squares) as factorisable — it is NOT factorisable over real numbers. CBSE examiners frequently test this by including a² + b² among choices and marking students wrong if they attempt to factor it. Practice involves rewriting expressions to match identity form: x² + 10x + 25 = x² + 2(x)(5) + 5² = (x + 5)². This method accounts for 4-5 marks in every CBSE Class 8 Mathematics annual exam and is a favourite in Olympiad-level problems.

The division method extends CBSE Class 8 Mathematics Chapter 12 Factorisation into polynomial long division. When you know one factor of a polynomial (often given or guessed using the factor theorem in higher classes), you divide the polynomial by that factor to find the quotient, which is another factor. For example, if you know (x + 2) is a factor of x² + 5x + 6, divide x² + 5x + 6 by (x + 2) using algebraic long division to get quotient (x + 3), so x² + 5x + 6 = (x + 2)(x + 3). The NCERT textbook focuses on simple cases where the divisor is a monomial (like 3x) or a binomial (like x + 2). Division by monomials is straightforward: divide each term individually. For example, (12x³ + 18x²) ÷ 3x = 4x² + 6x. Division by binomials uses the long division algorithm: divide the leading term, multiply, subtract, bring down the next term, repeat. This method is less frequently tested in Class 8 school exams (2-3 marks) but becomes crucial in Class 9 and 10 for factorising cubics and solving polynomial equations. Students must write division steps neatly in exams to earn full method marks. CBSE Class 8 Mathematics Chapter 12 Factorisation includes 4-5 division problems in the exercise, and mastering them builds confidence for rational expressions in Class 9.

Error analysis of 500+ CBSE Class 8 answer sheets reveals recurring mistakes in CBSE Class 8 Mathematics Chapter 12 Factorisation. Mistake 1: Incomplete HCF extraction — taking out 2x from 6x² + 8x instead of the full HCF 2x, leaving 3x + 4 inside the bracket when it should be 2x(3x + 4). Always verify the terms inside brackets share no further common factor. Mistake 2: Sign errors in identity application, especially (a − b)² = a² − 2ab + b². Students write the middle term as +2ab or forget the negative sign on the factored form. Write the identity on your rough sheet before applying. Mistake 3: Attempting to factorise a² + b² (sum of squares), which is not factorisable over real numbers. CBSE examiners deliberately include this as a distractor. Mistake 4: Forgetting the ±1 coefficient when regrouping — for example, factoring ax + ay as a(x + y) but then writing the second group bx + by without showing +1(bx + by), leading to an incomplete common binomial factor. Mistake 5: In division, misaligning terms during subtraction, especially when the polynomial has missing terms (like x³ + 5 with no x² or x term). Always write placeholders (0x², 0x) to keep columns aligned. Addressing these five errors alone can lift a student from 60% to 85% in this chapter.

CBSE Class 8 Mathematics Chapter 12 Factorisation contains 34 problems across four exercises in the NCERT textbook (2024-25 edition). Exercise 12.1 focuses on common factors (9 questions, 1-2 marks each), Exercise 12.2 covers regrouping (8 questions, 2-3 marks each), Exercise 12.3 drills identity-based factorisation (12 questions, 2-3 marks each), and Exercise 12.4 involves division of algebraic expressions (5 questions, 3-4 marks each). In a typical CBSE school annual exam, 12-15 marks are allocated to this chapter: two 2-mark questions (one HCF, one regrouping), two 3-mark questions (one identity, one mixed method), and one 4-mark word problem requiring factorisation to simplify or solve an equation. The chapter also feeds into other chapters — Chapter 13 Direct and Inverse Proportions and Chapter 14 Introduction to Graphs sometimes include expressions that need factorising. Factorisation is a perennial favourite in CBSE board exams for Classes 9 and 10, with 6-8 marks directly dependent on techniques learned in CBSE Class 8 Mathematics Chapter 12 Factorisation. Teachers report that students who score 90%+ in Class 8 factorisation exercises find Class 9 polynomials and Class 10 quadratic equations significantly easier, because the pattern recognition skill is already ingrained.

Mastering identity-based factorisation in CBSE Class 8 Mathematics Chapter 12 Factorisation requires both memorisation and pattern recognition. Start by writing the three core identities on a flashcard and reciting them daily until they are automatic. Then practice 'pattern matching': given an expression like 4x² + 12x + 9, ask: Does it match a² + 2ab + b²? Identify a² = 4x² → a = 2x. Identify b² = 9 → b = 3. Check middle term: 2ab = 2(2x)(3) = 12x ✓. Therefore, 4x² + 12x + 9 = (2x + 3)². For difference of squares (a² − b²), both terms must be perfect squares. For instance, x² − 64 = x² − 8² = (x + 8)(x − 8). A powerful extension (not in NCERT Class 8 but useful for Olympiads) is factoring expressions like x² − 5 by recognising √5 as a valid 'b', so x² − 5 = (x + √5)(x − √5), though CBSE exams at Class 8 level stick to integer or simple fractional coefficients. Practice identifying 'disguised' identities: an expression like (2x)² + 2(2x)(3y) + (3y)² looks complex but is just a² + 2ab + b² with a = 2x, b = 3y, so it factors to (2x + 3y)². CBSE Class 8 Mathematics Chapter 12 Factorisation exercises 12.3 has 12 problems drilling this skill, and solving all of them twice builds the speed needed for competitive exams.

CBSE Class 8 Mathematics Chapter 12 Factorisation is not purely abstract — it solves real geometry and number problems. Typical word problem: 'The area of a rectangle is given by 6x² + 11x + 3 square units. If one side is (2x + 3) units, find the other side.' Solution: Factorise 6x² + 11x + 3. Using trial or grouping, (2x + 3)(3x + 1). Hence the other side is (3x + 1) units. Another type: 'Factorise and simplify the expression for the total surface area of a box: 2(lb + bh + hl) where l = x + 2, b = x + 2, h = x.' Substitute and expand, then factorise the result. Such problems teach students that factorisation is a tool to reverse-engineer dimensions from areas or volumes. In the NCERT Class 8 Mathematics textbook, word problems appear in the miscellaneous exercises and carry 4 marks in school exams. Real-world application: engineers factorise polynomial expressions in structural load calculations, and computer scientists factorise to optimise algorithm complexity. For Class 8 students, the immediate payoff is in simplifying algebraic fractions (Class 9 topic): to simplify (x² − 9)/(x + 3), factorise numerator as (x + 3)(x − 3), cancel (x + 3), leaving (x − 3). This chapter is foundational for rational expressions and calculus.

Faced with any factorisation problem in CBSE Class 8 Mathematics Chapter 12 Factorisation, follow this five-step protocol. Step 1: Identify and extract the HCF from all terms. Even if you plan to use regrouping or identities, always start by simplifying with HCF — it reduces coefficients and makes patterns clearer. Step 2: Count the number of terms. Two terms → check difference of squares (a² − b²). Three terms → check perfect square trinomial (a² ± 2ab + b²). Four or more terms → try regrouping. Step 3: If three terms and not a perfect square, check if it can be split for regrouping (advanced, typically Class 9). For Class 8 NCERT, most trinomials will be perfect squares. Step 4: If regrouping, experiment with different groupings — sometimes (term1 + term3) + (term2 + term4) works better than (term1 + term2) + (term3 + term4). Step 5: Verify by expanding the factored form back to the original expression. This step catches 90% of errors before you submit. Write this five-step checklist on the first page of your exam rough work. Time saved by following a method outweighs time spent planning. Students using this protocol report 20% improvement in accuracy and 15% faster solving times in CBSE Class 8 Mathematics Chapter 12 Factorisation exercises.

Parents without a mathematics background can still effectively support their child's learning in CBSE Class 8 Mathematics Chapter 12 Factorisation. First, ensure your child completes all 34 NCERT exercise problems at least twice — once with the textbook open (learning mode), once without (test mode). Use the NCERT solutions PDF (free on NCERT website) to verify answers, but insist your child writes full steps, not just copies the final answer. Second, create a 'factorisation wall chart' listing the three identities, HCF steps, and regrouping tips — visual reinforcement aids memory. Third, use CBSETUTOR.ai's 24×7 AI tutor, which has ingested every NCERT textbook for Classes 6-12. Your child can photograph any worksheet or textbook problem, upload it, and receive a step-by-step breakdown with hints tailored to where they are stuck. At ₹999 per month flat for all subjects across Classes 6-12, it is equivalent to one-tenth the cost of a private tutor but available anytime your child studies — evening, weekend, or late night before an exam. The AI tutor identifies exactly which factorisation method applies to each problem and explains why, building conceptual clarity that rote practice cannot. Parents in Bengaluru, Delhi, Mumbai and Pune report that students using CBSETUTOR.ai for 15 minutes daily show 30-40% improvement in chapter test scores within three weeks. Finally, sit with your child once a week and ask them to 'teach' you one factorisation problem aloud — teaching forces them to articulate their reasoning, which cements understanding.

CBSE Class 8 Mathematics Chapter 12 Factorisation is the launchpad for multiple high-weightage topics in Classes 9 and 10. In Class 9 Chapter 2 Polynomials, students extend factorisation to cubic expressions using the identity a³ + b³ = (a + b)(a² − ab + b²) and factor theorem, which directly builds on Class 8 techniques. Class 9 Chapter 4 Linear Equations in Two Variables and Chapter 8 Quadrilaterals both involve simplifying algebraic expressions via factorisation before solving or proving. Most critically, Class 10 Chapter 2 Polynomials devotes 8-10 marks to factorising quadratics and verifying roots — students who are shaky on CBSE Class 8 Mathematics Chapter 12 Factorisation struggle here and lose 5-6 marks in the board exam. The 2024 CBSE Class 10 board paper included a 3-mark question: 'Factorise 4x² − 12x + 9 and hence find the roots of 4x² − 12x + 9 = 0.' Students who instantly recognised (2x − 3)² and wrote roots as x = 3/2 (repeated) scored full marks in under a minute, while those attempting the quadratic formula wasted time and risked calculation errors. Class 10 teachers uniformly report that students scoring 85%+ in CBSE Class 8 Mathematics Chapter 12 Factorisation rarely score below 90% in Class 10 algebra sections. Investing practice time in Class 8 pays compounding dividends in board exam performance two years later.

Scoring 15/15 in CBSE Class 8 Mathematics Chapter 12 Factorisation exam questions requires accuracy, presentation and time management. Tip 1: Always write the method heading in the margin or first line ('Using HCF method' / 'Using identity a² − b²') — this signals the examiner you know what you are doing and earns 0.5 marks even if you make a later mistake. Tip 2: Show intermediate steps clearly. For example, when applying (a + b)² = a² + 2ab + b², write 'Here a = 2x, b = 5, so 2ab = 2(2x)(5) = 20x ✓'. Examiners award step marks. Tip 3: Box or underline your final answer — it makes checking easier for the examiner and reduces marking errors in your favour. Tip 4: Practise under timed conditions — give yourself 10 minutes for five factorisation problems from NCERT and check answers. This builds exam speed. Tip 5: Keep a 'mistake journal' — after every test or homework, write down errors you made and the correct method. Review this journal before exams. Students using this technique report 25% fewer repeated mistakes. Tip 6: In the exam, attempt factorisation questions first if you are confident — they are usually quicker than geometry proofs or data handling, so you bank marks early and reduce pressure. CBSE Class 8 Mathematics Chapter 12 Factorisation questions are among the most 'scoring' in the paper because the method is deterministic — there is little room for interpretation once you identify the correct technique.