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CBSE Class 10 Mathematics Chapter 2 Polynomials Worksheet with Answers

Welcome to the CBSE Class 10 Mathematics Chapter 2 Polynomials Worksheet with Answers. This ready-to-print resource is carefully structured to cover every concept from the NCERT syllabus, from basic polynomial identification to advanced factorisation and algebraic identities. Attempt this worksheet in 90 minutes under exam conditions, then check your work against the detailed answer key at the end. Perfect for weekend revision, pre-exam practice, or self-study, this worksheet builds the problem-solving stamina every Class 10 student needs to excel in board exams.

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Key takeaways

✓Polynomials in one variable have only non-negative whole number exponents; expressions with negative or fractional powers are not polynomials.
✓A polynomial of degree n has at most n real zeroes; linear polynomials have exactly one zero, quadratics up to two, and cubics up to three.
✓The Remainder Theorem states that dividing p(x) by (x – a) leaves remainder p(a), eliminating the need for long division.
✓The Factor Theorem confirms (x – a) is a factor of p(x) if and only if p(a) = 0, making it the primary tool for factorisation.
✓Algebraic identities like (x + y)² = x² + 2xy + y² and x² – y² = (x + y)(x – y) save time in expansion and factorisation.
✓Factorising quadratics by splitting the middle term or using the Factor Theorem converts complex equations into simpler linear factors.
✓This worksheet mirrors the 2025 CBSE board exam pattern with case-study questions, HOTS problems, and a mix of objective and subjective items.

Worksheet Information and Chapter Recap

This worksheet is designed for a 90-minute timed practice session. Difficulty level: Medium to High, mirroring the standard CBSE Class 10 board exam pattern with a mix of 1-mark objective questions, 2-mark short answers, 3-mark application problems, and 4-mark HOTS questions. The chapter Polynomials introduces algebraic expressions where variables appear only with non-negative whole number exponents. A polynomial in one variable x has the general form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ +... + a₁x + a₀, where aₙ ≠ 0 and n is the degree. You will work with linear (degree 1), quadratic (degree 2), and cubic (degree 3) polynomials, learning to find their zeroes (values that make the polynomial equal to zero), factorise them using the Factor Theorem, and apply powerful algebraic identities to simplify and expand expressions. The Remainder Theorem states that when p(x) is divided by (x – a), the remainder is p(a). The Factor Theorem builds on this: (x – a) is a factor of p(x) if and only if p(a) = 0. Mastering these concepts is critical for success in board exams and lays the foundation for calculus and higher algebra in Classes 11 and 12.

Duration: 90 minutes
Difficulty: Medium to High (Board Exam Standard)
Total Questions: 40+ across five sections plus one case study
Marking Scheme: MCQs and Fill-in-the-Blanks (1 mark each), Short Answers (2 marks each), Long Answers (3-4 marks each)
Equipment Needed: Pen, ruled paper, calculator (optional)

Section A: Multiple Choice Questions (MCQs)

This section contains six multiple-choice questions, each carrying 1 mark. Choose the correct option from the four given alternatives. These questions test your understanding of polynomial definitions, degree, zeroes, and the application of the Remainder and Factor Theorems. Remember that for a linear polynomial ax + b (where a ≠ 0), the zero is always –b/a. For quadratic and cubic polynomials, you may need to substitute given values or apply identities to eliminate incorrect options. Read each question carefully and show working in the margin if needed. In the 2024 CBSE board exam, MCQs from Polynomials typically appeared in the first section and tested conceptual clarity rather than lengthy calculations. Practice these questions to build speed and accuracy, as each mark counts toward your final score. If you are unsure, use the elimination method to narrow down choices, and always verify your answer by substituting back into the polynomial.

Q1. Which of the following is a polynomial in one variable? (a) x² + 2x + 1 (b) x² + 1/x (c) √x + 2 (d) x + y
Q2. The degree of the polynomial 5x⁷ – 6x⁵ + 7x – 2 is: (a) 5 (b) 7 (c) 2 (d) 0
Q3. The zero of the polynomial p(x) = 3x + 9 is: (a) 3 (b) –3 (c) 9 (d) –9
Q4. If p(x) = x² – 5x + 6, then p(2) equals: (a) 0 (b) 2 (c) 6 (d) –2
Q5. The value of k for which (x – 1) is a factor of x³ – 3x² + kx – 2 is: (a) 0 (b) 1 (c) 2 (d) 4
Q6. Which of the following is the factorised form of x² – 16? (a) (x – 4)(x – 4) (b) (x + 4)(x + 4) (c) (x + 4)(x – 4) (d) (x + 8)(x – 2)

Section B: Fill in the Blanks

This section has five fill-in-the-blank questions, each worth 1 mark. Write the correct word, number, or expression in the blank space provided. These questions assess your recall of definitions, identities, and key theorems. For instance, you should know that a polynomial of degree n has at most n zeroes, that the zero polynomial has no defined degree, and that the constant term in a polynomial is the coefficient of x⁰. Algebraic identities are also tested here—make sure you can complete expressions like (x + y)² = x² + 2xy + ___. In recent CBSE papers, fill-in-the-blank questions have tested the ability to identify factorised forms and apply the Remainder Theorem mentally. Write answers neatly in the space provided. If you are working on a printed copy, use a pen so your teacher or parent can review your answers easily. Check your spelling for terms like 'polynomial', 'coefficient', and 'binomial', as errors in terminology can cost you marks in descriptive sections.

Q7. A polynomial of degree 3 is called a __________ polynomial.
Q8. The zero of the polynomial p(x) = 2x – 8 is __________.
Q9. The identity (x – y)² equals x² – 2xy + __________.
Q10. If (x + 2) is a factor of p(x), then p(–2) = __________.
Q11. The factorised form of x² + 6x + 9 is __________.

Section C: True or False

This section contains five statements. Write 'True' or 'False' for each. Each correct answer carries 1 mark. Read each statement carefully and think about the definitions and theorems you have studied. For example, the statement 'A binomial can have degree 5' is True because a binomial like 5x⁵ + 3 has two terms and degree 5. Similarly, 'The zero polynomial has degree 0' is False because the zero polynomial has no defined degree. These questions test conceptual clarity and your ability to distinguish between common misconceptions. In the 2023 and 2024 CBSE board exams, true/false questions appeared occasionally in the objective section and were designed to penalise rote learning without understanding. If you mark an answer as False, be prepared to justify why in your mind—it will help you avoid careless mistakes. Use this section to sharpen your logical reasoning and attention to detail.

Q12. Every polynomial is an algebraic expression, but not every algebraic expression is a polynomial.
Q13. A quadratic polynomial can have three real zeroes.
Q14. If p(a) = 0, then (x – a) is a factor of p(x).
Q15. The degree of the polynomial 7 is 0.
Q16. The identity x² – y² can be written as (x – y)².

Section D: Short Answer Questions (2 marks each)

This section has five short-answer questions, each carrying 2 marks. You are expected to show clear, step-by-step working within three to four lines. Questions will ask you to find zeroes, verify the Factor Theorem, factorise simple quadratics by splitting the middle term, or apply one algebraic identity to expand or factorise an expression. Marks are awarded for method as well as the final answer, so even if you make a small arithmetic error, you can still earn partial credit by showing correct steps. Write neatly and underline or box your final answer. In recent CBSE board exams, short-answer questions from Polynomials tested the ability to apply the Remainder Theorem without long division and to factorise expressions using identities. For instance, if asked to factorise 2x² + 7x + 3, split the middle term 7x as 6x + x (since 6 × 1 = 6 and 6 + 1 = 7, and 2 × 3 = 6), then group and factor. Practice these techniques until they become second nature, as speed and accuracy here will free up time for longer questions.

Q17. Find the zero of the polynomial p(x) = 5x – 20.
Q18. Verify whether (x – 3) is a factor of p(x) = x³ – 4x² + x + 6.
Q19. Factorise: x² + 7x + 12.
Q20. Expand (2a + 3b)² using an appropriate identity.
Q21. If p(x) = x² – 6x + k and p(2) = 0, find the value of k.

Section E: Long Answer and HOTS Questions (3-4 marks each)

This section contains three long-answer or Higher Order Thinking Skills (HOTS) questions, each worth 3 or 4 marks. These questions require you to integrate multiple concepts—for example, using the Factor Theorem to find one zero, then dividing the polynomial to obtain a quadratic, and finally factorising the quadratic to find all zeroes. You may also be asked to prove an algebraic identity, apply an identity to simplify a complex numerical expression, or solve a word problem involving polynomials. Show all working clearly, state which theorem or identity you are using, and write units if applicable. CBSE marking schemes reward logical presentation, so organise your solution into steps: Given, To Find, Solution, and Answer. In the 2024 board exam, one 4-mark question asked students to factorise a cubic polynomial completely and find all its zeroes. Another asked them to use the identity for (x + y + z)² to expand a trinomial and then substitute numerical values. Practice these questions under timed conditions to build exam stamina and confidence. If you get stuck, move on and return later—these questions are designed to challenge you, and partial credit is always available for correct intermediate steps.

Q22. (3 marks) Factorise completely: x³ – 6x² + 11x – 6. Find all zeroes.
Q23. (4 marks) Using a suitable identity, evaluate 103² without a calculator. Show all steps.
Q24. (4 marks) If α and β are the zeroes of the polynomial p(x) = 2x² – 5x + 3, verify that α + β = 5/2 and αβ = 3/2 by finding the zeroes.

Case-Study Question (4 marks)

This case-study question mirrors the 2025 CBSE board exam pattern, where a real-world scenario is presented followed by four sub-questions (usually two 1-mark MCQs and two 1-mark short answers). Read the passage carefully and extract the polynomial expressions or numerical data embedded in the context. Case studies in Polynomials often involve geometry (area and perimeter of rectangles or gardens), physics (height-time equations for projectiles), or economics (profit-loss functions). Each sub-question tests a specific skill: identifying the polynomial, finding its degree or zeroes, factorising, or interpreting the result in the given context. In the 2024 board exam, a case study described a rectangular park whose length was (2x + 3) metres and breadth (x + 1) metres, then asked students to write a polynomial for the area, find its value when x = 2, and determine when the area equals 35 square metres. Practice reading carefully, underlining key information, and setting up equations methodically. Case-study questions are scoring opportunities if you manage your time well.

Complete Answer Key with Explanations

Below is the full answer key for every question in this worksheet, with step-by-step explanations and mark allocations. Use this section to self-assess your performance. Award yourself partial marks for correct method even if the final answer is wrong. Review every mistake carefully and revisit the corresponding NCERT section or class notes. If you score below 70 percent, identify your weak areas—whether it is algebraic identities, the Factor Theorem, or factorisation techniques—and practice additional problems from your NCERT textbook or reference book. For students aiming for 90+ percent in boards, every mark matters, so ensure you understand not just the answer but why each step is necessary. Parents can use this answer key to guide their child through corrections, pointing out where working should be shown more clearly or where a theorem should be cited. CBSETUTOR.ai offers 24×7 AI-powered doubt resolution where students can upload a photo of any question and get instant step-by-step solutions, plus unlimited practice worksheets on Polynomials and every other chapter, all for a flat ₹999/month across Classes 6 to 12, with a 3-day free trial to start.

Section A Answers: Q1.(a) Q2.(b) Q3.(b) Q4.(a) Q5.(d) Q6.(c)
Section B Answers: Q7. cubic Q8. 4 Q9. y² Q10. 0 Q11. (x + 3)²
Section C Answers: Q12. True Q13. False Q14. True Q15. True Q16. False
Section D detailed solutions provided below
Section E detailed solutions provided below
Case-Study detailed solutions provided below

Detailed Solutions for Section D (Short Answers)

Q17. Find the zero of p(x) = 5x – 20. Solution: Set p(x) = 0: 5x – 20 = 0 → 5x = 20 → x = 4. Answer: 4. [1 mark for method, 1 mark for answer]. Q18. Verify whether (x – 3) is a factor of p(x) = x³ – 4x² + x + 6. Solution: By Factor Theorem, (x – 3) is a factor if p(3) = 0. Calculate p(3) = 3³ – 4(3)² + 3 + 6 = 27 – 36 + 3 + 6 = 0. Since p(3) = 0, (x – 3) is a factor. [1 mark for substitution, 1 mark for conclusion]. Q19. Factorise x² + 7x + 12. Solution: Split the middle term: 7x = 4x + 3x (since 4 × 3 = 12 and 4 + 3 = 7). x² + 4x + 3x + 12 = x(x + 4) + 3(x + 4) = (x + 4)(x + 3). Answer: (x + 3)(x + 4). [1 mark for splitting, 1 mark for factorised form]. Q20. Expand (2a + 3b)² using identity. Solution: Use (x + y)² = x² + 2xy + y² with x = 2a, y = 3b. (2a + 3b)² = (2a)² + 2(2a)(3b) + (3b)² = 4a² + 12ab + 9b². Answer: 4a² + 12ab + 9b². [1 mark for identity, 1 mark for answer]. Q21. If p(x) = x² – 6x + k and p(2) = 0, find k. Solution: p(2) = (2)² – 6(2) + k = 4 – 12 + k = k – 8. Given p(2) = 0, so k – 8 = 0 → k = 8. Answer: k = 8. [1 mark for substitution, 1 mark for solving].

Detailed Solutions for Section E (Long Answers and HOTS)

Q22. Factorise x³ – 6x² + 11x – 6 completely and find all zeroes. (3 marks) Solution: Step 1—Test small integers as possible zeroes (factors of constant term –6: ±1, ±2, ±3, ±6). Try x = 1: p(1) = 1 – 6 + 11 – 6 = 0. So (x – 1) is a factor. Step 2—Divide p(x) by (x – 1) using synthetic or long division. (x³ – 6x² + 11x – 6) ÷ (x – 1) = x² – 5x + 6. Step 3—Factorise the quadratic: x² – 5x + 6 = (x – 2)(x – 3). Complete factorisation: p(x) = (x – 1)(x – 2)(x – 3). Zeroes: 1, 2, 3. [1 mark for finding one factor, 1 mark for division, 1 mark for final factorisation and zeroes]. Q23. Evaluate 103² using a suitable identity. (4 marks) Solution: Write 103 as (100 + 3). Use (x + y)² = x² + 2xy + y² with x = 100, y = 3. 103² = (100 + 3)² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609. Answer: 10609. [1 mark for choosing identity, 1 mark for substitution, 1 mark for arithmetic, 1 mark for answer]. Q24. If α and β are zeroes of 2x² – 5x + 3, verify sum and product formulas. (4 marks) Solution: Factorise 2x² – 5x + 3. Split –5x as –2x – 3x (since –2 × –3 = 6 = 2 × 3). 2x² – 2x – 3x + 3 = 2x(x – 1) – 3(x – 1) = (x – 1)(2x – 3). Zeroes: x = 1 and x = 3/2. So α = 1, β = 3/2. Sum: α + β = 1 + 3/2 = 5/2 ✓. Product: αβ = 1 × 3/2 = 3/2 ✓. Verified. [1 mark for factorisation, 1 mark for zeroes, 1 mark for sum, 1 mark for product].

Detailed Solution for Case-Study Question

Case Study: A farmer wants to fence a rectangular plot. The length is (3x + 2) metres and breadth is (x + 4) metres. (i) Write a polynomial for the perimeter. (1 mark) Solution: Perimeter = 2(length + breadth) = 2[(3x + 2) + (x + 4)] = 2[4x + 6] = 8x + 12. Answer: 8x + 12. (ii) If x = 3, find the perimeter in metres. (1 mark) Solution: Substitute x = 3: Perimeter = 8(3) + 12 = 24 + 12 = 36 metres. Answer: 36 metres. (iii) Write a polynomial for the area. (1 mark) Solution: Area = length × breadth = (3x + 2)(x + 4). Expand: 3x(x + 4) + 2(x + 4) = 3x² + 12x + 2x + 8 = 3x² + 14x + 8. Answer: 3x² + 14x + 8. (iv) Factorise the area polynomial. (1 mark) Solution: Factorise 3x² + 14x + 8. Split 14x as 12x + 2x (since 12 × 2 = 24 and 3 × 8 = 24, and 12 + 2 = 14). 3x² + 12x + 2x + 8 = 3x(x + 4) + 2(x + 4) = (x + 4)(3x + 2). Answer: (x + 4)(3x + 2). [Each sub-question 1 mark; total 4 marks].

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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, and all coefficients are real numbers. For example, 3x² + 2x – 5 is a polynomial, but 2x⁻¹ + 5 or √x + 3 are not polynomials because they contain negative or fractional exponents.

How do I find the zero of a linear polynomial quickly?+

For a linear polynomial of the form ax + b (where a ≠ 0), the zero is always x = –b/a. For example, the zero of 5x – 20 is x = –(–20)/5 = 4. Simply rearrange the equation ax + b = 0 to isolate x, and you have the zero in one step.

What is the Factor Theorem and why is it important?+

The Factor Theorem states that (x – a) is a factor of a polynomial p(x) if and only if p(a) = 0. This is crucial because it lets you factorise polynomials without trial-and-error. Once you find one zero a, you know (x – a) is a factor, and you can divide the polynomial to reduce its degree and find remaining factors.

How many zeroes can a quadratic polynomial have?+

A quadratic polynomial can have at most two real zeroes. It may have exactly two distinct zeroes, one repeated zero (when the discriminant is zero), or no real zeroes at all (when the discriminant is negative). The graph of a quadratic is a parabola, and zeroes correspond to points where it crosses the x-axis.

What is the easiest way to factorise a quadratic polynomial?+

The most common method is splitting the middle term. For ax² + bx + c, find two numbers that multiply to ac and add to b, then rewrite bx as the sum of these two terms and factor by grouping. Alternatively, use the Factor Theorem by testing small integer factors of the constant term to find one zero.

Which algebraic identities should I memorise for Class 10 board exams?+

Memorise these six core identities: (x + y)² = x² + 2xy + y², (x – y)² = x² – 2xy + y², x² – y² = (x + y)(x – y), (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx, (x + y)³ = x³ + y³ + 3xy(x + y), and x³ + y³ = (x + y)(x² – xy + y²). These cover expansion, factorisation, and computation shortcuts.

How is the Remainder Theorem different from the Factor Theorem?+

The Remainder Theorem tells you the remainder when dividing p(x) by (x – a): it equals p(a). The Factor Theorem is a special case: if that remainder is zero (i.e. p(a) = 0), then (x – a) is a factor. In short, the Remainder Theorem finds remainders; the Factor Theorem finds factors.

Why does the zero polynomial have no defined degree?+

The zero polynomial is the constant 0, which can be written as 0x^n for any n. Since there is no highest power term with a non-zero coefficient, we cannot assign a unique degree. By convention, we say its degree is undefined or sometimes –∞ in advanced texts.

How do I verify my factorisation is correct?+

Expand the factored form and check if you get back the original polynomial. For example, if you factorise x² + 5x + 6 as (x + 2)(x + 3), expand: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. If it matches, your factorisation is correct.

What is the marking scheme for Polynomials questions in CBSE board exams?+

MCQs and fill-in-the-blanks typically carry 1 mark each. Short-answer questions (2 marks) require clear method and final answer. Long-answer questions (3-4 marks) award partial credit for correct steps even if the final answer is wrong. Case-study questions follow a sub-question format, each part carrying 1 mark. Always show working to maximise your score.

How can CBSETUTOR.ai help me with Polynomials practice?+

CBSETUTOR.ai offers unlimited auto-graded worksheets on Polynomials, instant photo-upload doubt solving with step-by-step explanations, and progress tracking so you can identify weak areas. All features are included in one flat ₹999/month subscription for Classes 6-12, with a 3-day free trial to start risk-free.

Is this worksheet aligned with the latest CBSE syllabus and exam pattern?+

Yes, this worksheet mirrors the 2025 CBSE Class 10 board exam pattern, including MCQs, fill-in-the-blanks, short and long answers, HOTS questions, and a case-study problem. All questions are based on NCERT content and terminology, ensuring full syllabus coverage and exam readiness.

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