Class 9 Mathematics Chapter 2 Polynomials Previous Year Questions: Complete PYQ Bank with Solutions

Reading your NCERT textbook and re-reading solutions is passive learning. Solving past papers forces active recall, time management, and pattern recognition—the three pillars of exam success. When you solve a PYQ on 'find the zeros of 3x² + x − 4', you must recall the factorization method, apply it under timed conditions, and check your answer. This builds automaticity. Additionally, CBSE examiners repeat certain question types because they test fundamental skills. For instance, 'If α and β are zeros of x² − 5x + 6, find α + β and αβ' appears in multiple years because it directly tests Vieta's formulas. By solving 5–10 similar questions, you internalize the relationship p(x) = x² − (sum of zeros)x + product of zeros. Theory alone cannot replicate this embedded learning. PYQs also reveal the exact language, notation, and diagram styles examiners use—matching these patterns in your answers improves marks. Studies on learning science confirm that retrieval practice (solving past papers) produces 50% better retention than re-reading. Start with 1-mark questions to warm up, move to 3-mark questions to build reasoning, then tackle 5-mark questions that combine multiple skills. This scaffolded approach, mirrored in Class 9 CBSE Mathematics previous year paper structures, ensures you're never overwhelmed.

One-mark questions are rapid-fire checks of vocabulary, formulas, and basic computation. Here are five frequently repeated patterns from past papers: **Q1: Degree Identification** "What is the degree of the polynomial 5x⁴ − 3x² + 2x − 7?" *Answer: 4* (The highest power of x is 4.) **Q2: Zeros Definition** "If p(x) = x² − 9, find the zeros of p(x)." *Answer: 3 and −3* (Solve x² − 9 = 0 → x = ±3.) **Q3: Coefficient Recognition** "In the polynomial 2x³ − 5x² + 0x + 8, what is the coefficient of x?" *Answer: 0* (The term containing x has coefficient 0.) **Q4: Vieta's Formula – Sum of Zeros** "If α and β are zeros of x² − 7x + 10, what is α + β?" *Answer: 7* (For ax² + bx + c, sum of zeros = −b/a = 7/1 = 7.) **Q5: Polynomial or Not** "Is 1/x + x² − 3 a polynomial? Why or why not?" *Answer: No.* (Polynomials cannot have negative integer exponents; 1/x = x⁻¹ disqualifies it.) **Why These Repeat:** Examiners test whether you've memorized definitions and can apply basic formulas instantly. Spending 2–3 minutes drilling 20 similar 1-mark questions builds confidence and secures guaranteed marks.

Three-mark questions require a short solution (3–5 logical steps) and test your ability to apply concepts. Here are five archetypal questions that have appeared repeatedly: **Q1: Find Zeros by Factorization** "Find the zeros of p(x) = x² + 5x + 6 and verify the relationship between zeros and coefficients." *Solution:* Factorize: x² + 5x + 6 = (x + 2)(x + 3). Zeros: α = −2, β = −3. Sum: α + β = −5 = −(coefficient of x)/(coefficient of x²) = −5/1 ✓. Product: αβ = 6 = (constant term)/(coefficient of x²) = 6/1 ✓. (3 marks: 1 for factorization, 1 for zeros, 1 for verification.) **Q2: Division Algorithm Application** "Divide p(x) = x³ + 2x² − 5x − 6 by g(x) = x − 2 and write the result as p(x) = g(x)q(x) + r(x)." *Solution:* Use long division or synthetic division. Quotient q(x) = x² + 4x + 3, remainder r = 0. Thus, x³ + 2x² − 5x − 6 = (x − 2)(x² + 4x + 3). (3 marks: 1 for setup, 1 for division steps, 1 for final form.) **Q3: Using Remainder Theorem** "If p(x) = 2x³ − 3x + 1, find the remainder when p(x) is divided by (x − 1) without division." *Solution:* By Remainder Theorem, remainder = p(1) = 2(1)³ − 3(1) + 1 = 2 − 3 + 1 = 0. (3 marks: 1 for stating theorem, 1 for substitution, 1 for answer.) **Q4: Zeros and Coefficients with Quadratic** "If the zeros of 2x² − bx + c are 2 and 3, find b and c." *Solution:* Sum of zeros = 2 + 3 = 5 = b/2 → b = 10. Product = 2 × 3 = 6 = c/2 → c = 12. (3 marks: 1 for applying sum formula, 1 for b, 1 for c.) **Q5: Polynomial Division with Remainder** "Divide x⁴ − 3x² + 2x + 5 by x² + 1 and express the result in the form p(x) = d(x)q(x) + r(x)." *Solution:* Using polynomial long division: quotient = x² − 4, remainder = 2x + 9. Thus x⁴ − 3x² + 2x + 5 = (x² + 1)(x² − 4) + (2x + 9). (3 marks: distributed across division accuracy and final form.) **Common Mistakes to Avoid:** Forgetting to verify zeros satisfy the sum–product relationship; making arithmetic errors in long division; confusing the remainder with the quotient.

Five-mark questions demand multi-step solutions combining 2–3 concepts. These are your high-scoring targets. **Q1: Find Zeros and Verify Factorization** "The polynomial p(x) = x³ − 6x² + 11x − 6 has one zero at x = 1. Find all three zeros and factorize p(x) completely. Verify using the division algorithm." *Solution:* Since p(1) = 1 − 6 + 11 − 6 = 0, (x − 1) is a factor. Divide p(x) by (x − 1) using long division: p(x) ÷ (x − 1) = x² − 5x + 6. Factorize x² − 5x + 6 = (x − 2)(x − 3). Thus p(x) = (x − 1)(x − 2)(x − 3). All zeros: 1, 2, 3. Verification: Sum = 1 + 2 + 3 = 6 (matches coefficient rule for x³ − 6x² + 11x − 6). Expand (x − 1)(x − 2)(x − 3) to confirm it equals p(x). (5 marks: 1 for identifying (x − 1), 1 for division, 1 for factorization of quotient, 1 for all three zeros, 1 for verification.) **Q2: Division Algorithm with Quotient and Remainder** "Divide x⁴ + x³ − 2x − 1 by x² − 1, express the result as p(x) = d(x)q(x) + r(x), and verify the division." *Solution:* Perform polynomial long division of x⁴ + x³ − 2x − 1 by x² − 1. First: x⁴ ÷ x² = x², multiply x²(x² − 1) = x⁴ − x², subtract from p(x). Continue: x³ + x² − 2x − 1 → next quotient term x, then 1. Final quotient: q(x) = x² + x + 1, remainder: r(x) = 2x. Thus x⁴ + x³ − 2x − 1 = (x² − 1)(x² + x + 1) + 2x. Verification: Expand (x² − 1)(x² + x + 1) + 2x and confirm = p(x). (5 marks: 2 for division steps, 1 for quotient, 1 for remainder, 1 for verification.) **Q3: Word Problem on Zeros and Quadratic Polynomial** "A rectangular garden has length 3 m more than its width. If the area is 40 m², form a quadratic equation, find the dimensions, and verify that the zeros satisfy Vieta's relationships." *Solution:* Let width = x m, length = (x + 3) m. Area = x(x + 3) = 40 → x² + 3x − 40 = 0. Factorize: (x + 8)(x − 5) = 0 → x = −8 or x = 5. Since width > 0, x = 5 m (width), 8 m (length). Verification: Area = 5 × 8 = 40 ✓. For quadratic x² + 3x − 40 = 0 with roots α = 5, β = −8: α + β = 5 + (−8) = −3 = −(coefficient of x)/(coefficient of x²) = −3/1 ✓. αβ = 5 × (−8) = −40 = constant/coefficient of x² = −40/1 ✓. (5 marks: 1 for forming equation, 1 for factorization/solving, 1 for interpreting context (rejecting negative), 1 for verification of Vieta's, 1 for clarity.) **Examination Strategy for 5-Mark Questions:** Allocate 12–15 minutes per question. Write every step clearly, even if intermediate. Always verify your answer using an alternative method (e.g., expand factored form to check against original polynomial). This reduces careless errors and shows examiner your reasoning.

The 2026–27 CBSE Mathematics curriculum maintains Chapter 2 Polynomials but with subtle emphasis shifts. The rationalized syllabus has increased focus on real-world applications and reduced rote memorization. Key observations from recent trends: **Shift 1: More Application-Based Questions** Expect fewer pure theoretical questions ('Define a polynomial') and more contextual problems ('A company's profit follows p(t) = −t² + 10t; when is profit zero?'). This mirrors the competency-based learning directive. **Shift 2: Division Algorithm Emphasis** The new pattern places greater weight on understanding division algorithm as a unifying principle. Questions now ask you to justify why p(x) = d(x)q(x) + r(x) and relate it to Remainder/Factor Theorem, rather than just performing division mechanically. **Shift 3: Graphical Understanding** Some papers now include questions like 'Sketch the graph of p(x) = x² − 5x + 6 and mark its zeros.' This bridges algebra and coordinate geometry—a cross-curricular expectation. **Shift 4: Higher-Order Thinking in 5-Mark** Five-mark questions increasingly ask 'If p(x) and g(x) have a common zero, prove that...' or 'Compare two polynomials in terms of their zeros.' These demand proof-writing and logical reasoning, not just computation. **Implication for Your Preparation:** Ensure you can (i) explain *why* division algorithm works, not just perform it; (ii) link zeros to factorization to graphs conceptually; (iii) write clear mathematical justifications for your steps. Practice explaining answers aloud—this signals deeper understanding to examiners. Start a 3-day free trial at cbsetutor.ai to access personalized drills aligned with the 2026–27 pattern.

Exam psychology matters as much as knowledge. Here's a battle-tested approach used by high-scoring students: **Pre-Exam Prep (1 Week Before)** Solve at least one full 5-mark question daily and three 3-mark questions. Time yourself: 3-mark should take ≤8 minutes, 5-mark ≤12 minutes. This builds speed without sacrificing accuracy. Keep a 'mistake log'—each time you err, write the error type (e.g., 'forgot negative sign in Vieta's formula') and the fix. **During Exam (Section A: 1-Mark Questions)** Allocate 10–12 minutes to 4 one-mark questions on Polynomials. Read each question twice. For 'find zeros of x² − 4', mentally factorize before writing. If unsure, move on—don't spend >90 seconds per 1-mark. These are meant to be quick confidence boosters. **During Exam (Section B: 3-Mark Questions)** For a 3-mark question, spend 30 seconds reading and planning. Write every step: (i) identify method (factorization, division, Remainder Theorem), (ii) execute, (iii) verify if applicable. Examiners award marks for method even if final answer has a small error. If you're stuck on factorization, switch to another method (e.g., use quadratic formula or trial-and-error with small integers). **During Exam (Section C: 5-Mark Questions)** These reward clarity. After reading, jot a quick outline: step 1 = divide by (x − 1), step 2 = factorize quotient, step 3 = list all zeros, step 4 = verify. Then execute. Box your final answer. If you realize you've made an error halfway, strike it out neatly, write 'CORRECTION:' and redo—examiners respect this honesty. **Red Flags to Avoid** (i) Not writing intermediate steps for division; (ii) Forgetting to verify zeros by substitution; (iii) Mixing up sum and product of zeros; (iv) Leaving remainder unanswered in division questions. **If You're Short on Time** Prioritize: 1-mark (attempt all), then 3-mark (attempt 2–3, fully worked), then 5-mark (attempt 1, fully worked). A partially solved 5-mark with clear reasoning scores more than a rushed, error-filled attempt at multiple 5-marks.

Success in Polynomials hinges on fluency with these core relationships. Memorize them and apply them mechanically in every relevant question. **1. Vieta's Formulas for Quadratic** If α and β are zeros of ax² + bx + c, then: α + β = −b/a (sum of zeros) αβ = c/a (product of zeros) Converse: If sum = S and product = P, the quadratic is x² − Sx + P. **2. Division Algorithm** For any polynomials p(x) and d(x) with d(x) ≠ 0: p(x) = d(x) · q(x) + r(x) where degree of r(x) < degree of d(x). **3. Remainder Theorem** When p(x) is divided by (x − a), the remainder = p(a). **4. Factor Theorem** (x − a) is a factor of p(x) if and only if p(a) = 0. **5. Degree Rules** degree of p(x) + q(x) ≤ max(degree of p(x), degree of q(x)) degree of p(x) × q(x) = degree of p(x) + degree of q(x) **6. Polynomial for Given Zeros** If zeros are α and β, the polynomial is k(x − α)(x − β) for any non-zero constant k. **Practice Tip:** Write these six formulas on a card and review daily for one week. Then close the card and reproduce them from memory. Repeat until automatic. On exam day, you'll apply them without thinking, freeing mental energy for harder reasoning steps.