Chapter 2 — Polynomials is a short, high-yield chapter in the current 2026-27 CBSE Class 10 Maths syllabus. Below are complete, worked solutions for the standard Exercise 2.1 problems on finding zeroes of quadratic polynomials and verifying the relationship between zeroes and coefficients.
NCERT Solutions for Class 10 Maths Chapter 2: Polynomials (Exercise 2.1)
Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² − 2x − 8
Factorising: x² − 4x + 2x − 8 = x(x−4) + 2(x−4) = (x−4)(x+2). Zeroes: x = 4, x = −2.
Sum of zeroes = 4 + (−2) = 2 = −(−2)/1 = −(coefficient of x)/(coefficient of x²) ✓
Product of zeroes = 4 × (−2) = −8 = (−8)/1 = (constant term)/(coefficient of x²) ✓
(ii) 4s² − 4s + 1
= (2s − 1)². Zeroes: s = 1/2, 1/2.
Sum = 1/2 + 1/2 = 1 = −(−4)/4 ✓
Product = 1/2 × 1/2 = 1/4 = 1/4 ✓
(iii) 6x² − 3 − 7x (rewrite as 6x² − 7x − 3)
= 6x² − 9x + 2x − 3 = 3x(2x−3) + 1(2x−3) = (3x+1)(2x−3). Zeroes: x = −1/3, 3/2.
Sum = −1/3 + 3/2 = 7/6 = −(−7)/6 ✓
Product = (−1/3)(3/2) = −1/2 = −3/6 ✓
(iv) 4u² + 8u
= 4u(u + 2). Zeroes: u = 0, −2.
Sum = 0 + (−2) = −2 = −8/4 ✓
Product = 0 × (−2) = 0 = 0/4 ✓
(v) t² − 15
Zeroes: t = √15, −√15.
Sum = 0 = −0/1 ✓
Product = −15 = −15/1 ✓
(vi) 3x² − x − 4
= 3x² − 4x + 3x − 4 = x(3x−4) + 1(3x−4) = (3x−4)(x+1). Zeroes: x = 4/3, −1.
Sum = 4/3 + (−1) = 1/3 = −(−1)/3 ✓
Product = (4/3)(−1) = −4/3 ✓
Q2. Find a quadratic polynomial for each pair of sum and product of zeroes given below. (Using the general form x² − (sum)x + (product), scaled to clear fractions where useful.)
(i) 1/4, −1 → x² − (1/4)x − 1, or equivalently 4x² − x − 4.
(ii) √2, 1/3 → x² − √2x + 1/3, or equivalently 3x² − 3√2x + 1.
(iii) 0, √5 → x² + √5.
(iv) 1, 1 → x² − x + 1.
(v) −1/4, 1/4 → x² + (1/4)x + 1/4, or equivalently 4x² + x + 1.
(vi) 4, 1 → x² − 4x + 1.
Why This Chapter Matters
Polynomials builds the algebraic foundation for Chapter 4 (Quadratic Equations) and reappears in Class 11 algebra. In board exams it typically contributes 2–4 marks via a direct zero-finding or coefficient-relationship question, and understanding the sum/product relationship (rather than memorising it) makes several later chapters, including Arithmetic Progressions, noticeably easier.
How to Use These Solutions
Work through each polynomial yourself first, then compare your factorisation steps with the ones above — exam markers award partial credit for correct method even if a small arithmetic slip changes the final answer. You can download the full chapter from our Class 10 Maths NCERT book page, or see every solved chapter so far on our Chapter 1: Real Numbers solutions.
Frequently Asked Questions
Are these solutions for the current 2026-27 NCERT syllabus?
Yes — this covers the rationalised Chapter 2 (Polynomials) Exercise 2.1 as currently prescribed by NCERT for the 2026-27 academic session.
Do I need to memorise the sum/product formulas?
It helps to know them, but understanding why they work (by expanding a(x−α)(x−β) and comparing coefficients) means you can rederive them instantly if you forget, which is safer under exam pressure than pure memorisation.
