NCERT Solutions for Class 10 Maths Chapter 2: Polynomials (2026–27)

These Class 10 Maths Chapter 2 solutions cover Polynomials from the latest NCERT textbook (Reprint 2026–27). Every question of Exercise 2.1 and Exercise 2.2 is reproduced verbatim and solved step by step, including finding zeroes by factorisation, verifying the relationship between zeroes and coefficients, and forming quadratic polynomials from a given sum and product of zeroes — all verified against the book’s answer key.

Class: 10 Subject: Mathematics Chapter: 2 – Polynomials Exercises: 2.1, 2.2 Topic: Zeroes & coefficients Session: 2026–27
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Chapter 2 Overview

Chapter 2 of Class 10 Maths, Polynomials, builds on the polynomials you met in Class IX. It recalls that the degree of a polynomial is the highest power of the variable, and classifies polynomials as linear (degree 1), quadratic (degree 2) and cubic (degree 3). The heart of the chapter is the idea of a zero of a polynomial — a value of x for which p(x) = 0 — and its geometrical meaning: the zeroes of p(x) are exactly the x-coordinates of the points where the graph of y = p(x) meets the x-axis. Finally it establishes the elegant relationship between the zeroes and the coefficients of quadratic (and cubic) polynomials, which the exercises below put into practice.

Key Concepts & Definitions

Polynomial: an expression of the form anxn + … + a1x + a0; the highest power of x is its degree.

Linear / quadratic / cubic: polynomials of degree 1, 2 and 3, of general form ax + b, ax2 + bx + c and ax3 + bx2 + cx + d respectively (a ≠ 0).

Value of a polynomial: p(k) is the number obtained by replacing x with k in p(x).

Zero of a polynomial: a real number k is a zero of p(x) if p(k) = 0. The zero of a linear polynomial ax + b is −b/a.

Geometrical meaning: the zeroes of p(x) are the x-coordinates of the points where the graph y = p(x) cuts the x-axis. A degree-n polynomial meets the x-axis in at most n points, so it has at most n zeroes.

Number of zeroes: a quadratic polynomial has at most 2 zeroes; a cubic has at most 3.

Important Formulas (Chapter 2)

Zero of a linear polynomial: zero of ax + b is −b/a = −(constant term)/(coefficient of x).

Quadratic — sum of zeroes: if α, β are zeroes of ax2 + bx + c, then α + β = −b/a = −(coefficient of x)/(coefficient of x2).

Quadratic — product of zeroes: αβ = c/a = (constant term)/(coefficient of x2).

Forming a quadratic from its zeroes: a quadratic with sum of zeroes S and product P is k[x2 − Sx + P], for any non-zero real k (take k = 1 for the simplest answer).

Cubic — relations: if α, β, γ are zeroes of ax3 + bx2 + cx + d, then α + β + γ = −b/a,   αβ + βγ + γα = c/a,   αβγ = −d/a.

Exercise 2.1 Solutions

Questions are reproduced verbatim from the NCERT textbook; the worked solutions are original and verified against the answers at the back of the book.

1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i), (ii), (iii), (iv), (v) and (vi) — six separate graphs of y = p(x).

SOLUTION The number of zeroes of p(x) equals the number of points where the graph of y = p(x) cuts (touches) the x-axis. We simply count those points in each graph of Fig. 2.10. (i) The curve does not meet the x-axis at any point → No zeroes (0 zeroes). (ii) The curve meets the x-axis at exactly one point → 1 zero. (iii) The curve meets the x-axis at three points → 3 zeroes. (iv) The curve meets the x-axis at two points → 2 zeroes. (v) The curve meets the x-axis at four points → 4 zeroes. (vi) The curve meets the x-axis at three points → 3 zeroes. (Figure-based question; answers read from Fig. 2.10 and confirmed by the NCERT answer key.)

Exercise 2.2 Solutions

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x2 − 2x − 8    (ii) 4s2 − 4s + 1    (iii) 6x2 − 3 − 7x (iv) 4u2 + 8u    (v) t2 − 15    (vi) 3x2 − x − 4

SOLUTION (i) x2 − 2x − 8. Factorise: x2 − 2x − 8 = x2 − 4x + 2x − 8 = x(x − 4) + 2(x − 4) = (x − 4)(x + 2). Zeroes: x = 4, x = −2. Check: here a = 1, b = −2, c = −8. Sum = 4 + (−2) = 2 = −b/a = −(−2)/1 = 2. ✓   Product = 4 × (−2) = −8 = c/a = −8/1. ✓ Zeroes: 4, −2. (ii) 4s2 − 4s + 1. = (2s − 1)2 = (2s − 1)(2s − 1). Zeroes: s = 1/2, 1/2 (equal). Check: a = 4, b = −4, c = 1. Sum = 1/2 + 1/2 = 1 = −b/a = 4/4 = 1. ✓   Product = 1/2 × 1/2 = 1/4 = c/a = 1/4. ✓ Zeroes: 1/2, 1/2. (iii) 6x2 − 3 − 7x, i.e. 6x2 − 7x − 3. Split −7x (product 6 × (−3) = −18): 6x2 − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3). Zeroes: x = −1/3, x = 3/2. Check: a = 6, b = −7, c = −3. Sum = −1/3 + 3/2 = (−2 + 9)/6 = 7/6 = −b/a = 7/6. ✓   Product = (−1/3)(3/2) = −1/2 = c/a = −3/6 = −1/2. ✓ Zeroes: −1/3, 3/2. (iv) 4u2 + 8u. = 4u(u + 2). Zeroes: u = 0, u = −2. Check: a = 4, b = 8, c = 0. Sum = 0 + (−2) = −2 = −b/a = −8/4 = −2. ✓   Product = 0 × (−2) = 0 = c/a = 0/4 = 0. ✓ Zeroes: −2, 0. (v) t2 − 15. = (t − √15)(t + √15). Zeroes: t = √15, t = −√15. Check: a = 1, b = 0, c = −15. Sum = √15 + (−√15) = 0 = −b/a = 0. ✓   Product = (√15)(−√15) = −15 = c/a = −15/1. ✓ Zeroes: −√15, √15. (vi) 3x2 − x − 4. Split −x (product 3 × (−4) = −12): 3x2 − 4x + 3x − 4 = x(3x − 4) + 1(3x − 4) = (3x − 4)(x + 1). Zeroes: x = 4/3, x = −1. Check: a = 3, b = −1, c = −4. Sum = 4/3 + (−1) = 1/3 = −b/a = 1/3. ✓   Product = (4/3)(−1) = −4/3 = c/a = −4/3. ✓ Zeroes: −1, 4/3.

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) 1/4, −1    (ii) √2, 1/3    (iii) 0, √5 (iv) 1, 1    (v) −1/4, 1/4    (vi) 4, 1

SOLUTION A quadratic with sum of zeroes S and product P is x2 − Sx + P (taking the constant k = 1). Multiply through to clear fractions to get the neatest integer-coefficient form. (i) S = 1/4, P = −1. x2 − (1/4)x + (−1) = x2 − (1/4)x − 1. Multiplying by 4: 4x2 − x − 4. (ii) S = √2, P = 1/3. x2 − √2 x + 1/3. Multiplying by 3: 3x2 − 3√2 x + 1. (iii) S = 0, P = √5. x2 − 0·x + √5 = x2 + √5. (iv) S = 1, P = 1. x2 − x + 1 = x2 − x + 1. (v) S = −1/4, P = 1/4. x2 − (−1/4)x + 1/4 = x2 + (1/4)x + 1/4. Multiplying by 4: 4x2 + x + 1. (vi) S = 4, P = 1. x2 − 4x + 1 = x2 − 4x + 1. (Any non-zero multiple of these polynomials is also a correct answer, since k can be any real number.)

Common Mistakes to Avoid

Watch out for these

  • Getting the sign wrong in α + β = −b/a — the sum is minus b/a, while the product is plus c/a.
  • Forgetting to rearrange a polynomial into standard ax2 + bx + c order before reading off a, b, c (e.g. 6x2 − 3 − 7x must first become 6x2 − 7x − 3).
  • When forming a quadratic from S and P, writing x2 + Sx + P instead of x2 − Sx + P.
  • Treating a value where the graph only touches the x-axis as two separate zeroes — it is one (repeated) zero, counted as a single point.
  • Confusing the number of zeroes with the degree; a degree-n polynomial has at most n zeroes, not always exactly n.
  • Leaving fractional coefficients in the final polynomial when an equivalent integer-coefficient form is cleaner (multiply by a common multiple).

Practice MCQs & Assertion–Reason

1. The zeroes of the polynomial x2 − 2x − 8 are:

(a) 2, 4    (b) −2, 4    (c) 2, −4    (d) −2, −4

2. If α and β are the zeroes of ax2 + bx + c, then α + β equals:

(a) b/a    (b) −b/a    (c) c/a    (d) −c/a

3. The product of the zeroes of 3x2 − x − 4 is:

(a) 4/3    (b) −4/3    (c) 1/3    (d) −1/3

4. A quadratic polynomial whose sum of zeroes is 0 and product is √5 is:

(a) x2 + √5    (b) x2 − √5    (c) x2 + √5 x    (d) √5 x2 + 1

5. The number of zeroes a polynomial of degree 3 can have, at most, is:

(a) 1    (b) 2    (c) 3    (d) 4

6. The zeroes of 4s2 − 4s + 1 are:

(a) 1/2, 1/2    (b) 1/2, −1/2    (c) 1, 1    (d) 2, 2

7. The zero of the linear polynomial 2x + 3 is:

(a) 3/2    (b) −3/2    (c) 2/3    (d) −2/3

8. The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x):

(a) meets the y-axis    (b) meets the x-axis    (c) has a maximum    (d) passes through the origin

9. The zeroes of t2 − 15 are:

(a) 15, −15    (b) √15 only    (c) √15, −√15    (d) no real zeroes

10. If one zero of 4u2 + 8u is 0, the other zero is:

(a) 2    (b) −2    (c) 8    (d) −8

Answer key: 1-(b), 2-(b), 3-(b), 4-(a), 5-(c), 6-(a), 7-(b), 8-(b), 9-(c), 10-(b).

For each Assertion–Reason question, choose: (A) Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion; (B) Both are true but the Reason is not the correct explanation; (C) Assertion is true but Reason is false; (D) Assertion is false but Reason is true.

A-R 1. Assertion: The zeroes of x2 − 2x − 8 are 4 and −2.

Reason: x2 − 2x − 8 factorises as (x − 4)(x + 2).

A-R 2. Assertion: For ax2 + bx + c the product of the zeroes is c/a.

Reason: For ax2 + bx + c the sum of the zeroes is −b/a.

A-R 3. Assertion: A quadratic polynomial can have at most two zeroes.

Reason: The graph of a quadratic polynomial meets the x-axis in at most two points.

A-R 4. Assertion: x2 + √5 is a quadratic polynomial whose sum of zeroes is 0 and product is √5.

Reason: A quadratic with sum of zeroes S and product P can be written as x2 − Sx + P.

A-R 5. Assertion: The polynomial 4s2 − 4s + 1 has two equal zeroes.

Reason: 4s2 − 4s + 1 is a perfect square, (2s − 1)2.

Answer key: 1-(A), 2-(B), 3-(A), 4-(A), 5-(A).

Quick Revision Summary

  • Polynomials of degree 1, 2 and 3 are linear, quadratic and cubic; a quadratic has the form ax2 + bx + c with a ≠ 0.
  • A real number k is a zero of p(x) if p(k) = 0; the zero of ax + b is −b/a.
  • The zeroes of p(x) are the x-coordinates of the points where y = p(x) meets the x-axis.
  • A quadratic has at most 2 zeroes; a cubic has at most 3 (degree n → at most n zeroes).
  • For a quadratic ax2 + bx + c: α + β = −b/a and αβ = c/a.
  • A quadratic with sum S and product P of zeroes is k[x2 − Sx + P].
  • For a cubic ax3 + bx2 + cx + d: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.

How to score full marks in this chapter

Always rewrite a quadratic in standard ax2 + bx + c order before factorising or reading off coefficients. Find zeroes by splitting the middle term, then show the verification — write α + β and αβ and compare with −b/a and c/a, because the relationship-check carries marks. When forming a polynomial from a given sum and product, write x2 − Sx + P, then multiply out fractions for a clean integer answer, and remember any non-zero multiple is acceptable.

Frequently Asked Questions

What is Class 10 Maths Chapter 2 Polynomials about?

Chapter 2 covers the degree of a polynomial, linear/quadratic/cubic polynomials, the zeroes of a polynomial and their geometrical meaning (where the graph meets the x-axis), and the relationship between the zeroes and the coefficients of quadratic and cubic polynomials.

How many exercises are there in Class 10 Maths Chapter 2?

In the NCERT 2026–27 textbook there are two exercises — Exercise 2.1 (number of zeroes from graphs) and Exercise 2.2 (finding zeroes, verifying the zero–coefficient relationship, and forming quadratics) — both fully solved on this page.

What is the relationship between zeroes and coefficients?

If α and β are the zeroes of ax2 + bx + c, then the sum α + β = −b/a and the product αβ = c/a. For a cubic ax3 + bx2 + cx + d the three relations are α+β+γ = −b/a, αβ+βγ+γα = c/a and αβγ = −d/a.

Are these Class 10 Maths Chapter 2 solutions free?

Yes. All solutions are free and follow the official NCERT Mathematics textbook for the 2026–27 session, with every answer verified against the book’s answer key.

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