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Important Questions: CBSE Class 10 Mathematics Chapter 2 Polynomials

Polynomials form the backbone of Class 10 algebra, contributing 6-8 marks across the CBSE board exam. Questions range from 1-mark MCQs testing definitions and zero-finding to 5-mark case-based problems requiring identity application and factorisation. This page organises 18 real exam-style questions by marks weightage, provides model answers with working steps, and explains exactly how CBSE frames questions from NCERT Chapter 2. Whether you're revising for term exams or final boards, this structured question bank—grounded in official NCERT terminology—will sharpen your problem-solving speed and accuracy.

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Key takeaways

✓Chapter 2 Polynomials typically carries 6-8 marks in CBSE Class 10 board exams across both terms, split between MCQs and descriptive questions.
✓Factor Theorem and Remainder Theorem are high-scoring areas; CBSE often asks students to find zeroes or verify factors using these theorems.
✓Algebraic identities like (x + y)² and x³ + y³ + z³ – 3xyz appear in 3-mark and 5-mark questions for expansion, factorisation, and numerical computation.
✓Case-based questions introduced from 2021 onwards test polynomial applications in geometry, profit functions, or motion, requiring interpretation and multi-step working.
✓Common errors include sign mistakes in identity application, skipping verification steps in Factor Theorem, and incorrect handling of zero polynomial degree.
✓Practising 15-20 varied questions from each marks category ensures full coverage of NCERT exercise problems and additional CBSE sample-paper patterns.

Chapter Overview and Marks Weightage in CBSE Exams

Chapter 2 Polynomials is a core algebra chapter in CBSE Class 10 Mathematics, typically accounting for 6-8 marks in the annual board exam. According to the CBSE marking scheme released for 2024 and 2025 sample papers, questions are distributed as follows: 1-2 MCQs worth 1 mark each (testing definitions, degree, zeroes of linear polynomials), one 2-mark question on Remainder or Factor Theorem application, one 3-mark question on factorisation or identity expansion, and sometimes a 4-5 mark case-based question linking polynomials to real-life scenarios like profit functions or geometric area calculations. The chapter draws heavily from NCERT exercises 2.1 through 2.4, which cover polynomial identification, zero-finding, division algorithm, and algebraic identities. Students must master both computational skills (factorising cubics, expanding trinomials) and conceptual clarity (why the Factor Theorem works, how to verify zeroes). The difficulty level ranges from straightforward MCQs to multi-step problems requiring synthesis of Remainder Theorem, splitting the middle term, and identity recognition. Scoring full marks demands neat algebraic working, correct use of NCERT notation (p(x) for polynomial, α/β for zeroes), and verification steps. CBSE has consistently included one polynomial question in every term paper since the competency-based pattern began in 2021, making this chapter non-negotiable for board aspirants.

Typical weightage: 6-8 marks across MCQs (1-2 questions), VSA/SA (2-3 marks), and LA/case-based (4-5 marks).
High-scoring topics: Factor Theorem, algebraic identities (especially x² – y² and (x + y + z)²), and factorisation of quadratic/cubic polynomials.
NCERT exercises 2.2 (zeroes) and 2.4 (identities) are most frequently referenced in board questions.
Case-based questions (introduced 2021) test interpretation: e.g., 'A company's profit p(x) = x³ – 6x² + 11x – 6; find break-even points (zeroes)'.

One-Mark Questions: MCQs and Very Short Answer

One-mark questions test foundational concepts: polynomial identification, degree, number of zeroes, and basic identity application. CBSE typically includes 1-2 MCQs from this chapter in Section A of the board paper. These questions demand quick recall and conceptual clarity. For instance, recognising that x + 1/x is not a polynomial (negative exponent) or that a quadratic polynomial can have at most two zeroes. Students often lose marks by confusing degree with number of terms or by misidentifying the zero polynomial's degree (which is undefined, not zero). Practise these to ensure you claim easy marks in the exam hall.

Two-Mark Questions: Remainder and Factor Theorem Applications

Two-mark questions typically ask you to apply the Remainder Theorem (find remainder when p(x) is divided by x – a by computing p(a)) or the Factor Theorem (verify if x – a is a factor by checking if p(a) = 0). CBSE values clear substitution steps and correct arithmetic. A common error is sign mistakes when substituting negative numbers; for instance, if a = –2 and you have x³, students write (–2)³ = 8 instead of –8. Always show each step: 'Let p(x) = …, then p(a) = … = … = final value.' If the question asks 'Hence find the other factor,' use division or factorisation after confirming the first factor. These questions are easy marks if you write neatly and verify your answer.

Three-Mark Questions: Factorisation and Identity Application

Three-mark questions demand multi-step working: factorise a cubic polynomial by first finding one zero (trial-and-error with factors of the constant term), then dividing to obtain a quadratic, which you factorise by splitting the middle term. Alternatively, expand expressions using identities like (x + y + z)² or factorise using x² – y² or x³ + y³ + z³ – 3xyz. CBSE awards 1 mark for method, 1 for correct intermediate steps, and 1 for final answer. Show all substitutions clearly. For example, if factorising x³ – 6x² + 11x – 6, test x = 1: p(1) = 1 – 6 + 11 – 6 = 0, so (x – 1) is a factor. Divide to get x² – 5x + 6 = (x – 2)(x – 3). Final answer: (x – 1)(x – 2)(x – 3). Always verify by expanding back or checking zeroes. Identity questions might ask you to evaluate 103² using (100 + 3)² or to prove an equality for all x by expanding both sides.

Five-Mark and Case-Based Questions

Five-mark questions and case-based problems (introduced in CBSE 2021 pattern) integrate multiple concepts: polynomial division, zero-sum properties, and real-world applications. A typical case presents a scenario—profit function p(x) for x units sold, area polynomial for a geometric shape, or a motion equation—and asks you to find zeroes (break-even points, dimensions, time instants), verify factors, and interpret results. CBSE expects you to read the case carefully, extract the polynomial, apply Factor Theorem or identities, solve, and write a concluding statement linking back to the context. Marks are awarded for comprehension (1), correct method (2), computation (1), and conclusion (1). Always define variables, write 'Let p(x) = …', show substitution, and state the final answer in words: 'The company breaks even at 2 units and 3 units.' These questions test both mathematical skill and ability to communicate reasoning, mirroring real-world problem-solving.

How CBSE Frames Questions from Polynomials Chapter

CBSE question-setters follow predictable patterns rooted in NCERT exercises and learning outcomes. For Polynomials, they test five competencies: (1) Identifying polynomials and computing degree (MCQs from Ex 2.1). (2) Finding zeroes of linear and quadratic polynomials using formulae or factorisation (Ex 2.2 Q1-5). (3) Applying Remainder and Factor Theorems to verify factors or find unknown coefficients (Ex 2.3 Q1-3). (4) Factorising quadratics by splitting middle term and cubics by finding one zero then dividing (Ex 2.3 Q4-5). (5) Using algebraic identities for expansion, factorisation, and numerical computation (Ex 2.4 and 2.5). Since 2021, CBSE adds case-based questions where a polynomial models a real situation—profit, area, speed—and students must interpret, factorise, and conclude. The mark distribution mirrors Bloom's taxonomy: 1-mark MCQs test recall (knowledge), 2-mark questions test application (Remainder Theorem), 3-mark questions test analysis (multi-step factorisation), and 5-mark questions test synthesis and evaluation (case interpretation). CBSE rarely asks rote identity statements ('Write identity IV'); instead, they ask 'Expand (3a – 2b + c)² using an identity' or 'Factorise 64x³ – 125y³ using a suitable identity.' Understanding this pattern lets you predict question types and allocate study time: spend 40 per cent on factorisation, 30 per cent on identities, 20 per cent on theorems, 10 per cent on definitions.

CBSE never asks 'Define polynomial'—they ask 'Is √x + 2 a polynomial? Justify.' (application, not recall).
Remainder Theorem questions always specify a linear divisor (x – a) and expect substitution, not long division.
Factorisation questions on cubics give polynomials with integer zeroes (usually ±1, ±2, ±3) to keep arithmetic simple.
Identity-based questions prefer three-variable expansions or sum-of-cubes because they test deeper pattern recognition.
Case-based questions provide all data; no external knowledge needed—just read carefully and apply chapter concepts.

Common Mistakes Students Make and How to Avoid Them

Analysing CBSE answer scripts reveals recurring errors that cost students 2-4 marks per paper. (1) Sign errors in substitution: When finding p(–2) for p(x) = x³ + 2x² – x + 3, students write (–2)³ = 8 instead of –8, leading to wrong remainder. Always write intermediate steps: (–2)³ = –8, (–2)² = +4. (2) Forgetting to verify factors: After factorising x³ – 6x² + 11x – 6 as (x – 1)(x – 2)(x – 3), expand mentally or test one zero to confirm. CBSE awards method marks only if final answer is correct. (3) Misapplying identities: Using (x – y)² = x² – y² (wrong) instead of x² – 2xy + y². Write the full identity first, then substitute. (4) Incomplete factorisation: Writing 2x² + 5x + 3 = (2x + 3)(x + 1) and stopping, without checking if 2x + 3 can be factored further. Always factorise until all factors are linear or irreducible. (5) Dividing incorrectly: When dividing x³ – 2x² – x + 2 by (x – 1), students lose track of signs during long division. Use the division algorithm carefully: dividend = divisor × quotient + remainder. (6) Zero polynomial degree: Stating degree of the zero polynomial as 0 instead of 'undefined.' NCERT explicitly says it has no defined degree. (7) Not writing units or context in case-based answers: 'Zeroes are 1, 2, 4' instead of 'Company breaks even at 1, 2, 4 batches.' CBSE marking schemes reward contextual conclusions with 1 mark. Practice these mistakes in mock tests, self-correct, and maintain an error log to track patterns in your own working.

Sign errors: Always compute powers of negative numbers separately before substitution.
Identity confusion: Memorise all seven identities with correct signs; revise before each test.
Incomplete factorisation: After splitting middle term, check if each factor is prime or can be factored further.
Verification skipped: Substitute one zero back into original polynomial to confirm working.
Contextual statements: In case-based questions, end with a sentence linking zeroes to the real-world scenario.

Additional Practice Questions (3-Mark and 5-Mark)

Here are four more challenging questions to round out your practice. These mirror the difficulty level of CBSE board papers and sample papers released in 2024. Work through each step-by-step, referring back to NCERT worked examples if you get stuck. Time yourself: aim for 3 minutes per 3-mark question and 6 minutes per 5-mark question to build exam speed. After solving, check your working against the model answers and note any conceptual gaps. Regular timed practice with a variety of question types is the single best predictor of board exam performance in Polynomials.

Using CBSETUTOR.ai for Polynomials Mastery

Many Class 10 students in metro cities struggle with Polynomials because coaching classes rush through factorisation techniques without building conceptual clarity on why the Factor Theorem works or how identities derive from binomial expansion. Parents in Delhi-NCR, Mumbai, Bengaluru, and Tier-2 cities often spend ₹15,000-25,000 per month on tuition, yet students still make sign errors and skip verification steps because they lack personalised, on-demand doubt-solving. CBSETUTOR.ai offers a smarter solution: a 24×7 AI tutor that lets your child photograph any Polynomials problem—whether from NCERT, RD Sharma, or a mock test—and receive step-by-step worked solutions in seconds. The platform covers every Class 10 Maths chapter with NCERT-aligned explanations, practice question banks, and instant feedback on mistakes. At a flat ₹999 per month for all subjects across Classes 6-12, it costs less than two days of private tuition yet provides unlimited doubt-clearing. Start with a 3-day free trial: your child can upload yesterday's homework on algebraic identities, see the AI break down each substitution, and learn error-free technique before the next class test. For Polynomials specifically, CBSETUTOR.ai's question bank includes 50+ graded problems, video walkthroughs of every NCERT exercise, and a mistake-tracker that highlights recurring errors like sign slips or incomplete factorisation. This targeted practice builds both speed and accuracy, translating directly into board exam marks.

Photo-upload solving: Snap any Polynomials question; get worked solution with NCERT terminology in under 10 seconds.
Flat ₹999/month for Classes 6-12, all subjects—one price, no hidden fees, no per-question charges.
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Revision Strategy for Maximum Marks in Polynomials

With two weeks before your CBSE board exam, follow this proven revision plan for Polynomials: (Day 1-3) Re-read NCERT sections 2.1-2.5, underlining definitions (polynomial, degree, zero, factor) and all seven identities. Solve NCERT exercises 2.2, 2.3, 2.4 completely, checking answers against the back of the book. (Day 4-6) Attempt 10 previous years' board questions (2019-2024) under timed conditions—3 minutes per 2-3 mark question, 6 minutes per 5-mark question. Mark your answers against CBSE marking schemes available on cbse.nic.in; note which steps earn marks. (Day 7-9) Focus on your weak areas: if factorisation is slow, do 15 cubic factorisations using the 'find-one-zero-then-divide' method; if identities confuse you, write out each identity ten times and solve 5 application problems per identity. (Day 10-12) Solve two full CBSE sample papers (2024, 2025) and analyse errors. Create a one-page formula sheet with all identities, Remainder Theorem statement, and factorisation checklist. (Day 13-14) Rapid revision: read your formula sheet, solve 5 mixed questions (one per type), and visualise yourself writing neat, step-by-step solutions in the exam hall. On exam day, read Polynomials questions twice, identify the theorem or identity needed, write the formula first, then substitute. Allocate 15-18 minutes for Polynomials questions in a 3-hour paper, leaving time to verify answers. This disciplined approach, combined with NCERT exercise mastery, typically yields 6-7 out of 8 marks even for average students.

Memorise all seven algebraic identities verbatim; write them on the first page of your answer sheet if allowed.
For every factorisation, verify by expanding or testing one zero—CBSE awards full marks only for correct final answers.
Practise writing solutions in the exact format: 'Let p(x) = …', 'By Factor Theorem, p(a) = …', 'Hence the factors are …'
In case-based questions, underline key data in the problem, define the polynomial, solve, then write a concluding sentence.
Use CBSE marking schemes (available on official website) to understand how examiners award step-marks.

Frequently asked questions

How many marks does Polynomials carry in the CBSE Class 10 board exam?+

Chapter 2 Polynomials typically contributes 6-8 marks in the CBSE Class 10 annual board exam. This includes 1-2 MCQs (1 mark each), one 2-mark question on Remainder or Factor Theorem, one 3-mark factorisation or identity question, and sometimes a 4-5 mark case-based problem. The exact distribution varies slightly year to year, but CBSE consistently includes polynomial questions in both terms.

What are the most important topics in Polynomials for board exams?+

Focus on Factor Theorem applications, factorisation of quadratic and cubic polynomials by splitting the middle term or finding one zero, and algebraic identities—especially (x ± y)², x² – y², (x + y + z)², and x³ + y³ + z³ – 3xyz. CBSE frequently tests these in 3-mark and 5-mark questions. Additionally, practise Remainder Theorem for 2-mark questions and case-based interpretation for 5-mark problems.

How do I factorise a cubic polynomial quickly?+

First, find one zero by testing small integers that are factors of the constant term (try ±1, ±2, ±3, etc.). Once you find a zero c, write (x – c) as one factor. Then divide the cubic by (x – c) using long division or synthetic division to get a quadratic. Factorise the quadratic by splitting the middle term or using the formula. Verify by expanding all factors back to the original polynomial.

What is the difference between Remainder Theorem and Factor Theorem?+

Remainder Theorem states that when polynomial p(x) is divided by (x – a), the remainder equals p(a). Factor Theorem is a special case: (x – a) is a factor of p(x) if and only if p(a) = 0. In other words, if the remainder is zero, the divisor is a factor. Use Remainder Theorem to find remainders; use Factor Theorem to verify factors or find unknown coefficients.

Which algebraic identities should I memorise for Class 10 boards?+

Memorise these seven: (1) (x + y)² = x² + 2xy + y², (2) (x – y)² = x² – 2xy + y², (3) x² – y² = (x + y)(x – y), (4) (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx, (5) (x + y)³ = x³ + 3x²y + 3xy² + y³, (6) (x – y)³ = x³ – 3x²y + 3xy² – y³, (7) x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx). CBSE prefers identities 3, 4, and 7 in recent papers.

How many practice questions should I solve for Polynomials?+

Solve all NCERT exercises 2.1 through 2.5 (approximately 30 questions), then attempt 10-15 questions from each marks category (1-mark MCQs, 2-mark, 3-mark, 5-mark) from CBSE sample papers and previous years' papers (2019-2024). A total of 60-70 varied questions ensures you encounter every question pattern CBSE uses. Quality over quantity: review each solution, understand mistakes, and redo incorrect problems.

What are common mistakes in Polynomials that lose marks?+

Top errors: (1) Sign mistakes when substituting negative values (e.g. (–2)³ = 8 instead of –8). (2) Misapplying identities (using wrong formula or incorrect signs). (3) Incomplete factorisation (stopping at a factorable quadratic). (4) Forgetting to verify factors by substitution or expansion. (5) Not writing conclusion in case-based questions. (6) Stating degree of zero polynomial as 0 instead of undefined. Review CBSE marking schemes to see exactly where step-marks are awarded.

Is the zero polynomial's degree zero or undefined?+

The zero polynomial (the polynomial 0) has no defined degree according to NCERT. It is not assigned degree zero because every term is 0xⁿ for any n, so there is no highest power. CBSE has asked this as a 1-mark MCQ; the correct answer is 'undefined' or 'not defined,' not zero.

How do case-based questions on Polynomials work?+

CBSE case-based questions present a real-world scenario (company profit, geometric area, projectile height) modelled by a polynomial. You must: (1) Read the case and extract the polynomial. (2) Apply Remainder/Factor Theorem, find zeroes, or factorise as asked. (3) Interpret results in context (e.g. 'zeroes represent break-even points'). Marks: 1 for comprehension, 2 for method, 1 for computation, 1 for conclusion. Always end with a contextual statement linking your answer to the scenario.

Can I score full marks in Polynomials with only NCERT practice?+

Yes, if you solve every NCERT exercise problem thoroughly, understand the theory in each section, and practise writing neat, step-by-step solutions. However, supplement with 2-3 CBSE sample papers and 5-6 previous years' board questions to familiarise yourself with case-based formats and MCQ phrasing. NCERT exercises cover all concepts; sample papers add exam-pattern exposure. Together, they ensure 7-8 out of 8 marks in Polynomials.

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