India's #1 AI Tutorworksheet · Mathematics · Chapter 2

CBSE Class 9 Mathematics Chapter 2 Polynomials Worksheet with Answers

Polynomials form the backbone of algebra in CBSE Class 9 Mathematics. This chapter introduces you to expressions where variables appear with non-negative integer exponents, teaching you to classify, factorise, and solve them using powerful theorems and identities. This worksheet is structured to mirror the CBSE exam pattern with progressive difficulty levels, ensuring thorough practice across all question types.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓A polynomial has variables with only non-negative whole number exponents; expressions like x + 1/x are not polynomials.
✓The degree of a polynomial is the highest power of the variable with a non-zero coefficient; a degree-n polynomial has at most n zeroes.
✓The Factor Theorem states that (x – a) is a factor of p(x) if and only if p(a) = 0, making it essential for factorisation.
✓Algebraic identities like (x + y)² = x² + 2xy + y² and x² – y² = (x + y)(x – y) save time in expansion and factorisation.
✓The Remainder Theorem allows you to find the remainder of p(x) ÷ (x – a) simply by evaluating p(a), avoiding long division.
✓Splitting the middle term is the standard technique for factorising quadratic polynomials of the form ax² + bx + c.
✓Practice with varied question types—MCQs, fill-in-the-blanks, short and long answers—builds confidence for CBSE board exams.

Quick Chapter Recap: Polynomials at a Glance

A polynomial in one variable is an algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ +... + a₁x + a₀, where all exponents are non-negative whole numbers and coefficients are real. The degree is the highest power of x with a non-zero coefficient. Linear polynomials have degree 1, quadratic polynomials degree 2, and cubic polynomials degree 3. A zero of p(x) is a real number c such that p(c) = 0. The Remainder Theorem states that dividing p(x) by (x – a) yields remainder p(a). The Factor Theorem says (x – a) is a factor of p(x) if and only if p(a) = 0. Algebraic identities such as (x + y)² = x² + 2xy + y², x² – y² = (x + y)(x – y), and (x + y)³ = x³ + y³ + 3xy(x + y) are tools for quick expansion and factorisation. These concepts are tested in CBSE board exams through numerical problems, factorisation tasks, and application-based questions.

Polynomial: expression with non-negative integer exponents only
Degree: highest power of the variable with non-zero coefficient
Zeroes: values that make the polynomial equal to zero
Remainder Theorem: remainder when dividing p(x) by (x – a) is p(a)
Factor Theorem: (x – a) is a factor ⟺ p(a) = 0
Identities: memorise (x ± y)², x² – y², (x + y)³, x³ + y³ + z³ – 3xyz

Worksheet Instructions and Marking Scheme

This worksheet is designed for a 90-minute timed practice session and carries a total of 60 marks. Section A (MCQs) carries 1 mark each, Section B (fill-in-the-blanks) 1 mark each, Section C (match/true-false) 1 mark per correct item, Section D (short answers) 2 marks each, Section E (long answers and HOTS) 3 marks each, and the case study 4 marks. Attempt all questions in order. Use a pencil for diagrams and a pen for written answers. Show all working clearly for partial credit in Sections D and E. Calculators are not permitted. After completion, check your answers against the detailed answer key provided at the end. Mark yourself honestly and note topics where you lost marks for focused revision. This worksheet aligns with the latest CBSE Class 9 Mathematics syllabus and uses NCERT terminology throughout. For students seeking on-demand doubt clearance, CBSETUTOR.ai offers 24×7 AI-powered tutoring with photo-upload problem solving at ₹999/month for all classes 6–12, with a 3-day free trial.

Suggested time: 90 minutes
Total marks: 60
Difficulty level: Medium (suitable post-NCERT chapter reading)
No calculators; show all steps for partial marks
Self-assess using the answer key and identify weak areas

Section A: Multiple Choice Questions (1 mark each)

Choose the correct option for each question. Each MCQ tests a specific concept from the NCERT Class 9 Mathematics Chapter 2 syllabus. These questions range from definitions and degree identification to application of the Remainder and Factor Theorems and recognition of algebraic identities. Mark your answer clearly. No negative marking, so attempt all questions even if you are unsure. Refer to the answer key at the end for explanations of the correct choice and why the other options are incorrect. The questions below are representative of the types asked in CBSE board exams and school assessments.

Q1. Which of the following is a polynomial? (a) x² + 3x + 2 (b) x + 1/x (c) √x + 5 (d) x⁻³ + 2
Q2. The degree of the polynomial 5x⁴ – 3x² + 7 is: (a) 2 (b) 3 (c) 4 (d) 7
Q3. The zero of the polynomial p(x) = 2x – 6 is: (a) 2 (b) 3 (c) –3 (d) 6
Q4. If p(x) = x³ – 2x² + x – 1 is divided by (x – 1), the remainder is: (a) 0 (b) –1 (c) 1 (d) –3
Q5. Which of the following is a factor of x² – 5x + 6? (a) (x – 1) (b) (x – 2) (c) (x + 3) (d) (x + 2)
Q6. The value of (103)² using identity is: (a) 10506 (b) 10609 (c) 10209 (d) 10309
Q7. The factorised form of x² – 49 is: (a) (x – 7)(x – 7) (b) (x + 7)(x + 7) (c) (x + 7)(x – 7) (d) cannot be factorised
Q8. If p(2) = 0 for polynomial p(x), then one factor of p(x) is: (a) (x + 2) (b) (x – 2) (c) 2x (d) x²

Section B: Fill in the Blanks (1 mark each)

Complete each statement with the correct mathematical term, expression, or number. These questions test your recall of definitions, standard forms, and theorem statements from NCERT Class 9 Mathematics Chapter 2. Write your answer in the blank space provided. Be precise: for algebraic expressions, use correct notation and signs. Spelling and notation matter for technical terms like 'polynomial', 'coefficient', and 'degree'. The answer key provides the exact term or expression expected. This section builds confidence in terminology, which is essential for board exam theory questions and for clearly communicating solutions in long-answer problems.

Q9. A polynomial of degree 2 is called a __________ polynomial.
Q10. The coefficient of x in the polynomial 4x³ – 2x² + 5 is __________.
Q11. The value of k for which (x – 1) is a factor of x² – 3x + k is __________.
Q12. The identity (x + y)² is equal to __________.
Q13. If p(x) is divided by (x – a), then the remainder is __________ (by Remainder Theorem).
Q14. The number of zeroes a cubic polynomial can have at most is __________.
Q15. The factorised form of x² + 6x + 9 is __________.

Section C: Match the Following / True or False (1 mark each)

This section contains two parts. Part I: Match each polynomial expression in Column A with its correct property or factorised form in Column B. Write the correct letter in the answer column. Part II: State whether each statement is True or False. If false, you do not need to correct it here, but note it for revision. These questions assess your ability to quickly identify polynomial properties, apply identities, and recognize standard factorisations. Accuracy is key: carefully read each statement and match exactly. The answer key explains each match and the reasoning behind true or false judgments. Practising this format prepares you for objective-type questions common in CBSE school exams and Olympiad-level tests.

Part I – Match the Following:
Column A: (i) x² – 16 (ii) x² + 2x + 1 (iii) 2x + 8 (iv) x³ – 1
Column B: (a) (x + 1)² (b) (x – 1)(x² + x + 1) (c) (x – 4)(x + 4) (d) linear polynomial
Part II – True or False:
Q16. The polynomial x² + 1/x is a quadratic polynomial. (True / False)
Q17. Every linear polynomial has exactly one zero. (True / False)
Q18. The degree of the zero polynomial is 0. (True / False)

Section D: Short Answer Questions (2 marks each)

Answer each question in 2–3 sentences or show all necessary working. Each question is worth 2 marks: typically 1 mark for method and 1 mark for the correct final answer. Clearly state the theorem or identity you are using. For factorisation problems, show the splitting of the middle term or the test for zeroes. For evaluation problems, substitute carefully and simplify step-by-step. Partial marks are awarded for correct approach even if the final answer has a minor arithmetic error, so never leave a question blank. These questions mirror the short-answer section of CBSE board exams and are often drawn from NCERT exercise problems and examples. Use this section to practice writing concise, complete solutions under exam conditions. Time management tip: allocate about 3–4 minutes per question in this section during the actual worksheet attempt.

Q19. Find the zero of the polynomial p(x) = 5x – 20.
Q20. Using the Remainder Theorem, find the remainder when p(x) = x³ + 3x² – 5x + 4 is divided by (x – 2).
Q21. Factorise: x² – 7x + 12.
Q22. Expand using a suitable identity: (2a – 3b)².
Q23. Verify whether (x + 1) is a factor of x³ + x² – x – 1 using the Factor Theorem.

Section E: Long Answer and HOTS Questions (3 marks each)

These questions require multi-step solutions, application of multiple theorems, or higher-order thinking skills. Each is worth 3 marks: typically 1 mark for correct method, 1 mark for intermediate steps, and 1 mark for the final answer. Show all algebraic manipulation clearly. For factorisation of cubics, first find one zero by testing factors of the constant term, then perform polynomial division to reduce to a quadratic, and finally factorise the quadratic. For identity-based problems, choose the appropriate identity and substitute carefully. For proof or verification questions, state what you are proving and conclude clearly. These questions test deep understanding and problem-solving ability, preparing you for the toughest board exam questions. CBSETUTOR.ai helps students master such HOTS questions through step-by-step AI explanations and instant doubt resolution via photo upload, available 24×7 at just ₹999/month for classes 6–12 with a 3-day free trial. Allocate 6–7 minutes per question here during timed practice to ensure you finish the entire worksheet within 90 minutes.

Q24. Factorise completely: x³ – 6x² + 11x – 6.
Q25. Without actual multiplication, evaluate 98³ using a suitable algebraic identity.
Q26. If x + 1/x = 5, find the value of x² + 1/x².

Case-Study Based Question (4 marks)

Read the case study carefully and answer the sub-questions that follow. Case studies integrate real-life contexts with polynomial concepts and are now a regular feature of CBSE board exams. Marks are distributed across sub-parts, usually 1+1+2 or 1+1+1+1. Show working for all numerical sub-parts. This format tests comprehension, application, and analytical thinking. Practising case studies builds exam confidence and improves reading-comprehension skills under time pressure. The scenario below uses polynomials to model a practical situation, requiring you to extract information, form expressions, and solve equations. Carefully note the given data and what each sub-question asks before you begin writing your answer. Cross-check units and ensure your final answer makes sense in the real-world context described.

Case Study: A gardener designs a rectangular flower bed. The length of the bed is (x + 5) metres and the breadth is (x + 3) metres. The area of the bed is given by the polynomial A(x) = x² + 8x + 15.
Q27(i). Factorise the polynomial A(x). (1 mark)
Q27(ii). If the area of the flower bed is 35 square metres, form an equation and find the value of x. (2 marks)
Q27(iii). What are the actual length and breadth of the flower bed when the area is 35 m²? (1 mark)

Complete Answer Key with Explanations

This answer key provides the correct answer for every question along with concise explanations, key steps, or reasoning. Use it for self-assessment after completing the worksheet. Award yourself marks honestly: give full credit only when your method and answer both match, and partial credit when your approach is correct but you made an arithmetic slip. Note down question numbers where you lost marks and revisit those NCERT sections. For MCQs, understand why the correct option is right and why distractors are wrong. For theory questions, compare your phrasing with the model answer to improve exam writing. Regular practice with worksheets and honest self-evaluation using answer keys builds the discipline and accuracy needed to score high marks in CBSE board exams. If you find repeated mistakes in a particular topic—say, factorisation or identities—dedicate extra time to NCERT examples and exercises on that topic before attempting another worksheet. Digital platforms like CBSETUTOR.ai allow you to upload your worksheet answers via photo and receive instant AI feedback, helping you learn from errors immediately and clarify doubts 24×7 at a flat fee of ₹999/month across classes 6–12, with a 3-day free trial to get started.

Frequently asked questions

What is the difference between a polynomial and an algebraic expression?+

A polynomial is a special algebraic expression where the variable appears only with non-negative whole number exponents (0, 1, 2, 3,...) and all coefficients are real numbers. Expressions like x + 1/x or √x + 2 are algebraic but not polynomials because they contain negative or fractional exponents.

How do I quickly identify the degree of a polynomial?+

Look for the term with the highest power of the variable that has a non-zero coefficient. That exponent is the degree. For example, in 3x⁵ – 2x³ + 7, the degree is 5. Remember: the zero polynomial has no defined degree, and a non-zero constant like 5 has degree 0.

Why is the Factor Theorem so important for factorisation?+

The Factor Theorem tells you that if p(a) = 0, then (x – a) is a factor of p(x). This means you can test small integer values (factors of the constant term) to find one zero, immediately giving you one linear factor. Then divide the polynomial by that factor to reduce the degree and continue factorising.

Which algebraic identities should I memorise for Class 9 board exams?+

You must know (x + y)² = x² + 2xy + y², (x – y)² = x² – 2xy + y², x² – y² = (x + y)(x – y), (x + y)³ = x³ + y³ + 3xy(x + y), (x – y)³ = x³ – y³ – 3xy(x – y), and x³ + y³ + z³ – 3xyz = (x+y+z)(x²+y²+z²–xy–yz–zx). These save time in both expansion and factorisation.

How do I factorise a quadratic polynomial by splitting the middle term?+

For ax² + bx + c, find two numbers that multiply to a·c and add to b. Rewrite the middle term bx using these two numbers, then group and factor. For example, in 2x² + 7x + 3, find numbers that multiply to 6 and add to 7: these are 6 and 1. Write 2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (x+3)(2x+1).

What is the Remainder Theorem and when do I use it?+

The Remainder Theorem states that the remainder when polynomial p(x) is divided by (x – a) is simply p(a). Use it whenever you need the remainder without doing long division: just substitute a into the polynomial and evaluate. It is also the foundation of the Factor Theorem.

Can a polynomial have more zeroes than its degree?+

No. A polynomial of degree n can have at most n real zeroes. This is a fundamental result. For example, a quadratic (degree 2) can have 0, 1, or 2 zeroes, but never 3. A cubic polynomial can have up to 3 zeroes.

How can I use identities to compute squares and cubes mentally?+

Break the number into a sum or difference convenient for an identity. For 103², write it as (100 + 3)² = 100² + 2·100·3 + 3² = 10000 + 600 + 9 = 10609. For 97², use (100 – 3)² = 10000 – 600 + 9 = 9409. This technique is faster than manual multiplication and impresses examiners.

What are common mistakes students make in polynomial factorisation?+

Common errors include: sign mistakes when splitting the middle term, forgetting to check if the polynomial is fully factorised, and not verifying the factorisation by expanding. Always multiply your factors back to confirm you get the original polynomial, and watch plus/minus signs carefully.

How does CBSETUTOR.ai help with polynomials and other Class 9 Maths chapters?+

CBSETUTOR.ai provides 24×7 AI tutoring where you can upload photos of worksheet questions or doubts and receive instant step-by-step solutions and explanations. It covers all NCERT chapters for classes 6–12 at ₹999/month, with a 3-day free trial. This on-demand support helps clarify concepts like the Factor Theorem, factorisation techniques, and identity application anytime you are stuck, ensuring continuous learning without waiting for the next tuition class.

Related resources

Important Questions: CBSE Class 9 Mathematics Chapter 2 Polynomials Class 9 Mathematics Chapter 2 Polynomials — Formulas & Key Points NCERT Solutions for CBSE Class 9 Mathematics Chapter 2: Polynomials CBSE Class 9 Mathematics Chapter 2 Polynomials — Notes CBSE Class 9 Mathematics Chapter 1 Number Systems — Notes NCERT Solutions for CBSE Class 9 Mathematics Chapter 1: Number Systems NCERT Solutions for Class 9 Mathematics Chapter 2: Polynomials CBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

Ready to give your Class 9 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 9 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →