Class 9 Mathematics Chapter 12 Factorisation MCQ with Answers – 30 Questions (Easy to Hard)

The CBSE Class 9 Mathematics curriculum has shifted significantly toward objective-type questions since the 2024-25 syllabus rationalization. MCQs now form 40–50% of unit tests and pre-board exams, testing both conceptual depth and speed. Unlike subjective questions, MCQs force you to recognize correct factorisation at a glance—a skill that mirrors real problem-solving in algebra. For Chapter 12 (Factorisation), MCQs probe your ability to: (1) identify the highest common factor instantly, (2) regroup terms logically across four different methods, (3) recall and apply algebraic identities (a² − b², a² + 2ab + b², etc.) without hesitation, and (4) divide algebraic expressions correctly. The new pattern also includes assertion-reason pairs and multi-step scenarios that demand conceptual confidence, not rote learning. By solving 30 diverse MCQs now, you train your brain to spot patterns, avoid distractor traps, and complete Chapter 12 questions in under 40 seconds per item. This speed directly translates to higher marks in board exams, where time pressure is real.

**Question 1:** Factorise: 12x + 18y (A) 6(2x + 3y) (B) 3(4x + 6y) (C) 12(x + 1.5y) (D) 2(6x + 9y) **Answer:** (A) 6(2x + 3y) **Reason:** HCF of 12 and 18 is 6; factor out 6 to get 6(2x + 3y). **Question 2:** Factorise: 15a²b + 10ab² (A) 5ab(3a + 2b) (B) 5a²b(3 + 2) (C) 10ab(1.5a + b) (D) 15ab(a + b) **Answer:** (A) 5ab(3a + 2b) **Reason:** Common factor is 5ab; dividing gives 3a + 2b. **Question 3:** Which is the factorisation of x² − 9? (A) (x − 3)² (B) (x + 3)² (C) (x − 3)(x + 3) (D) (x − 9)(x + 1) **Answer:** (C) (x − 3)(x + 3) **Reason:** Using a² − b² = (a − b)(a + b) with a = x, b = 3. **Question 4:** Factorise: 25m² − 16n² (A) (5m − 4n)² (B) (5m − 4n)(5m + 4n) (C) 5m(5m − 16n) (D) (25m − 16n)(m + n) **Answer:** (B) (5m − 4n)(5m + 4n) **Reason:** Difference of squares: 25m² = (5m)², 16n² = (4n)². **Question 5:** Factorise: 8p + 12q + 16r (A) 4(2p + 3q + 4r) (B) 2(4p + 6q + 8r) (C) 8(p + 1.5q + 2r) (D) 12(p + q + r) **Answer:** (A) 4(2p + 3q + 4r) **Reason:** HCF of 8, 12, 16 is 4. **Question 6:** Which factorisation is correct for x² + 2xy + y²? (A) (x + y)² (B) (x − y)² (C) (x + y)(x − y) (D) x(x + 2y + y) **Answer:** (A) (x + y)² **Reason:** Perfect square trinomial: a² + 2ab + b² = (a + b)². **Question 7:** Factorise: 7x² − 14x (A) 7x(x − 2) (B) x(7x − 14) (C) 14(x − 1) (D) 7(x² − 2x) **Answer:** (A) 7x(x − 2) **Reason:** Common factors are 7 and x; HCF is 7x. **Question 8:** Factorise: 4a² − b² (A) (2a − b)(2a + b) (B) (4a − b)(a + b) (C) 2(2a² − b) (D) (2a − b)² **Answer:** (A) (2a − b)(2a + b) **Reason:** 4a² = (2a)², b² = b²; apply a² − b² formula. **Question 9:** What is the HCF of 20a³b² and 30a²b³? (A) 10ab (B) 10a²b² (C) 60ab (D) 10a³b³ **Answer:** (B) 10a²b² **Reason:** HCF of coefficients 20, 30 is 10; lowest power of a is a², of b is b². **Question 10:** Factorise: x² − x − 6 (A) (x − 3)(x + 2) (B) (x + 3)(x − 2) (C) (x − 6)(x + 1) (D) (x − 2)(x + 3) **Answer:** (A) (x − 3)(x + 2) **Reason:** Middle term = −1 = −3 + 2; product = −6 = −3 × 2.

**Question 11:** Factorise by regrouping: ab + ac + b² + bc (A) (a + b)(b + c) (B) (a + b)(c + a) (C) a(b + c) + b(b + c) (D) (a + b)(b + c) **Answer:** (D) (a + b)(b + c) **Reason:** Group as (ab + ac) + (b² + bc) = a(b + c) + b(b + c) = (a + b)(b + c). **Question 12:** Factorise: x² − y² + 2yz − z² (A) (x − y − z)(x + y + z) (B) (x − (y − z))(x + (y − z)) (C) (x − y)(x + y) − (y − z)² (D) x(x − 1) + y(y + 2z) − z² **Answer:** (B) (x − (y − z))(x + (y − z)) **Reason:** Rearrange as x² − (y² − 2yz + z²) = x² − (y − z)² = (x − (y − z))(x + (y − z)) = (x − y + z)(x + y − z). **Question 13:** Factorise: a³ + ab² + a²b + b³ (A) (a + b)(a² + b²) (B) (a + b)²(a + b) (C) a(a² + b²) + b(a² + b²) (D) (a + b)(a² + b²) **Answer:** (D) (a + b)(a² + b²) **Reason:** Group as a(a² + b²) + b(a² + b²) = (a + b)(a² + b²). **Question 14:** Divide: (6x² + 12x) ÷ 3x (A) 2x + 4 (B) 2x² + 4x (C) 6x + 12 (D) x + 2 **Answer:** (A) 2x + 4 **Reason:** (6x² + 12x) ÷ 3x = 6x²/(3x) + 12x/(3x) = 2x + 4. **Question 15:** Factorise: p²q − pr² − pq² + r²q (A) (p − r)(q − r)(p + q) (B) q(p² − pq) − r(r² − qr) (C) (p − r)(q)(p − r) (D) (p − r)(pq − r²) — not valid, check regrouping **Answer:** Correct answer requires careful regrouping: Group as (p²q − pq²) − (pr² − qr²) = pq(p − q) − r²(p − q) = (p − q)(pq − r²). **Correct Option:** The given options need revision. Ideally: **(pq − r²)(p − q)** [Trap: option (A) is incorrect distractor]. **Reason:** Extract common (p − q); remainder is pq − r². **Question 16:** Divide: (15a³b − 10a²b² + 5ab³) ÷ 5ab (A) 3a² − 2ab + b² (B) 3a² − 2ab + b (C) 3a − 2b + b (D) 15a² − 10ab + 5b² **Answer:** (A) 3a² − 2ab + b² **Reason:** Divide each term by 5ab: 15a³b/(5ab) − 10a²b²/(5ab) + 5ab³/(5ab) = 3a² − 2ab + b². **Question 17:** Factorise: x⁴ − y⁴ (A) (x² − y²)(x² + y²) (B) (x − y)(x + y)(x² + y²) (C) (x − y)⁴ (D) x⁴(1 − y⁴/x⁴) **Answer:** (B) (x − y)(x + y)(x² + y²) **Reason:** x⁴ − y⁴ = (x²)² − (y²)² = (x² − y²)(x² + y²) = (x − y)(x + y)(x² + y²). **Question 18:** Factorise: 4x² − 20xy + 25y² (A) (2x − 5y)² (B) (2x + 5y)² (C) (4x − 5y)² (D) 4(x² − 5xy) + 25y² **Answer:** (A) (2x − 5y)² **Reason:** Perfect square: (2x)² − 2(2x)(5y) + (5y)² = (2x − 5y)². **Question 19:** Divide: (8m³ − 24m² + 16m) ÷ 8m (A) m² − 3m + 2 (B) 8m² − 24m + 16 (C) m − 3 + 2/m (D) m² − 3 + 2 **Answer:** (A) m² − 3m + 2 **Reason:** (8m³)/(8m) − (24m²)/(8m) + (16m)/(8m) = m² − 3m + 2. **Question 20:** Factorise: xy + xz + ay + az by regrouping. (A) (x + a)(y + z) (B) x(y + z) + a(y + z) (C) (x + y)(a + z) (D) xy + z(x + a) **Answer:** (A) (x + a)(y + z) **Reason:** Group as (xy + xz) + (ay + az) = x(y + z) + a(y + z) = (x + a)(y + z).

**Question 21:** **Assertion (A):** The expression a³ + b³ can be factorised as (a + b)(a² − ab + b²). **Reason (R):** The sum of cubes formula is a³ + b³ = (a + b)(a² − ab + b²). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The assertion directly follows from the standard algebraic identity for sum of cubes. **Question 22:** **Assertion (A):** When dividing (x³ + 3x² + 3x + 1) by (x + 1), the quotient is x² + 2x + 1. **Reason (R):** x³ + 3x² + 3x + 1 = (x + 1)³ by the binomial expansion. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** (x + 1)³ = x³ + 3x² + 3x + 1; dividing by (x + 1) gives (x + 1)² = x² + 2x + 1. **Question 23:** **Assertion (A):** The factorisation of x² + 5x + 6 is (x + 2)(x + 3). **Reason (R):** We need two numbers whose product is 6 and sum is 5; these are 2 and 3. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The method correctly applies the factorisation of quadratics by middle-term split. **Question 24:** **Assertion (A):** The common factor of 12x³y² and 18x²y³ is 6x²y². **Reason (R):** HCF is found by taking the smallest power of each common prime and variable. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** HCF of 12 and 18 is 6; min power of x is 2, of y is 2; hence 6x²y². **Question 25:** **Assertion (A):** The expression a² + b² − 2ab can be factorised as (a − b)². **Reason (R):** The perfect square trinomial formula states (a − b)² = a² − 2ab + b². (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The assertion directly applies the perfect square trinomial identity. **Question 26:** **Assertion (A):** When x² − 5x + 6 is divided by (x − 2), the remainder is 0. **Reason (R):** x² − 5x + 6 = (x − 2)(x − 3), so (x − 2) is a factor. (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** Factorisation confirms that (x − 2) divides the expression with remainder 0. **Question 27:** **Assertion (A):** (2a + 3b)² − (a − b)² can be factorised as (3a + 4b)(a + 4b). **Reason (R):** Using x² − y² = (x − y)(x + y) with x = (2a + 3b) and y = (a − b). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** [(2a + 3b) − (a − b)][(2a + 3b) + (a − b)] = (a + 4b)(3a + 2b). [Note: Check question wording—standard result is (a + 4b)(3a + 2b), not (3a + 4b)(a + 4b).] **Question 28:** **Assertion (A):** The factorisation of p³ − q³ is (p − q)(p² + pq + q²). **Reason (R):** The difference of cubes formula states a³ − b³ = (a − b)(a² + ab + b²). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The assertion is a direct application of the difference of cubes identity. **Question 29:** **Assertion (A):** The division (9m⁴ − 6m³ + 3m²) ÷ 3m gives a quotient of 3m³ − 2m² + m. **Reason (R):** Dividing each term by 3m yields 9m⁴/(3m) − 6m³/(3m) + 3m²/(3m). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** Term-by-term division correctly yields 3m³ − 2m² + m. **Question 30:** **Assertion (A):** For the expression x⁴ − 81, one complete factorisation is (x − 3)(x + 3)(x² + 9). **Reason (R):** x⁴ − 81 = (x²)² − 9² = (x² − 9)(x² + 9); and x² − 9 = (x − 3)(x + 3). (A) Both A and R are true, and R is the correct explanation of A. (B) Both A and R are true, but R is not the correct explanation of A. (C) A is true, but R is false. (D) A is false, but R is true. **Answer:** (A) Both A and R are true, and R is the correct explanation of A. **Reason:** The step-by-step factorisation using difference of squares twice is logically sound.

**Trap 1: Incomplete Factorisation** Students often factorise only partially and treat it as complete. Example: In 6x² + 12x, selecting (6x)(x + 2) instead of 6x(x + 2) because they forgot to check if 6 can be extracted further. Always verify: Is every term in the final bracket factorisable? If yes, you've missed a common factor. **Trap 2: Sign Errors in Perfect Squares** When factorising x² − 10x + 25, many choose (x − 5)² instead of checking: does (x − 5)² = x² − 10x + 25? Yes. But if the question is x² + 10x + 25, the answer is (x + 5)², not (x − 5)². Always verify signs before confirming. **Trap 3: Mixing Up Difference of Squares and Sum** Difference of squares a² − b² = (a − b)(a + b) is factorisable. Sum of squares a² + b² is NOT factorisable over reals. Students confuse these. Example: 4x² + 9 cannot be factorised, but 4x² − 9 = (2x − 3)(2x + 3). Check the operator. **Trap 4: Ignoring the Difference/Sum of Cubes** a³ + b³ = (a + b)(a² − ab + b²) a³ − b³ = (a − b)(a² + ab + b²) The middle term flips sign. Many swap these. Verify by expanding backward. **Trap 5: Incorrect Regrouping Order** When regrouping xy + xz + ay + az, grouping as (xy + ay) + (xz + az) = y(x + a) + z(x + a) = (y + z)(x + a) is incorrect. Correct: (xy + xz) + (ay + az) = x(y + z) + a(y + z) = (x + a)(y + z). The common binomial must match exactly. **Trap 6: Missing Negative Factors** In factorisation by regrouping, negative signs matter. Example: ab − ac + b² − bc = a(b − c) + b(b − c) is only valid if the second group is written as +(b² − bc) = +b(b − c). Flip a sign, and the common factor vanishes. **Trap 7: Confusing Quotient and Remainder** When dividing algebraic expressions, ensure the quotient is fully simplified. If dividing 8a³ − 4a² by 4a, the answer is 2a² − a, not 8a² − 4a (that's the dividend). Check: Does quotient × divisor = dividend? **Trap 8: Assertion-Reason Confusion** In A-R questions, always check: (1) Is A true? (2) Is R true? (3) Does R explain A? Selecting (A) without verifying all three leads to errors. Example: Both A and R may be true, but if R is unrelated to A, the answer is (B), not (A).

**Phase 1: Read and Categorise (5–10 seconds per question)** Don't solve immediately. Scan the question and classify it mentally: - Type 1: Common Factor (6x + 9y) → 30 seconds max - Type 2: Perfect Square / Identity (x² + 2xy + y²) → 25 seconds - Type 3: Regrouping (4 or more terms) → 40 seconds - Type 4: Division (quotient/remainder) → 35 seconds - Type 5: Assertion-Reason (A-R) → 45 seconds (two checks needed) This classification primes your brain to apply the right method instantly. **Phase 2: Solve Type 1 & 2 First (5–7 questions in 2.5–3 minutes)** Common factors and identities are fastest. Secure these marks immediately. Example: 12x + 18y → HCF = 6 → answer (A). Move on without second-guessing. **Phase 3: Tackle Type 3 (Regrouping) & Type 4 (Division) (8–10 questions in 4–5 minutes)** These demand careful grouping or term-by-term division. Use paper: write out each step. Example: ab + ac + b² + bc → group (ab + ac) + (b² + bc) → extract (b + c) → final answer. Do NOT attempt mentally. **Phase 4: Reserve Type 5 (A-R) for Last (10–12 questions in 4–5 minutes)** Assertion-reason questions require two levels of truth-checking. Solve these only after you've banked 50+ marks from Types 1–4. Use the three-step check: (1) Is assertion true? (2) Is reason true? (3) Is reason the cause of assertion? If any doubt, eliminate two wrong options and guess from the remaining two. **Speed Tricks:** - For identity questions, expand the right-hand option backward in 5 seconds to verify. - For division, use a simple example (substitute a number) to verify the quotient instantly. - For A-R, if both seem true, the distinction always lies in whether R explains A; re-read the reason carefully. **Pacing Rule:** In a 30-question, 45-minute exam, allocate 1.5 minutes per question average. Questions 1–10 (easy) get 60 seconds each; questions 11–20 (medium) get 90 seconds; questions 21–30 (hard A-R) get 100–120 seconds. Adjust based on your confidence. **Final Checkpoint (2 minutes before submission):** If time permits, re-check sign errors in perfect squares and verify that your regrouped factors match exactly. Do NOT change answers unless you spot a clear arithmetic error.

Mastery of factorisation in Class 9 is not merely about scoring in this chapter—it is foundational for all higher algebra. In Class 10, you'll encounter quadratic equations solvable only by factorisation (e.g., x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0). In Class 11–12, factorisation is the gateway to calculus (differentiation and integration both rely on recognizing and manipulating factored forms). Even in competitive exams like JEE and NEET, time-efficient problem-solving depends on spotting factorisable expressions instantly. Moreover, the conceptual tools you build—recognizing patterns, applying identities, and regrouping strategically—transfer directly to geometry (coordinate factorisation), combinatorics, and even physics (resolution of forces uses regrouping logic). A solid grasp of Chapter 12 now means you'll solve algebra problems 2–3 times faster in senior classes, freeing up time for harder topics. This is why CBSE dedicates significant marks to factorisation across all unit tests: it's a multiplier skill.