Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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Complete NCERT Solutions for CBSE Class 10 Maths Chapter 2 — Polynomials, covering Exercise 2.1 (number of zeroes from graphs) and Exercise 2.2 (zeroes of quadratics and their relationship with coefficients).
Key Concepts (Quick Recap)
- A real number k is a zero of polynomial p(x) if p(k) = 0.
- A quadratic has at most 2 zeroes; a cubic has at most 3.
- Number of zeroes = number of times the graph meets the x-axis.
- For ax² + bx + c with zeroes α, β: α + β = −b/a, αβ = c/a.
- Polynomial from zeroes: x² − (α + β)x + αβ.
Exercise 2.1
Q1. The graphs of y = p(x) are given for some polynomials p(x). Find the number of zeroes of p(x) in each case.
Method: Count how many times each graph cuts or touches the x-axis — that count is the number of zeroes.
| Graph | Times it meets the x-axis | Number of zeroes |
|---|---|---|
| (i) | does not meet x-axis | 0 |
| (ii) | meets once | 1 |
| (iii) | meets three times | 3 |
| (iv) | meets twice | 2 |
| (v) | meets four times | 4 |
| (vi) | meets three times | 3 |
(Read each value directly from the figure in your textbook — the rule is always "count the x-axis intersections.")
Exercise 2.2
Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² − 2x − 8 Factorise: x² − 2x − 8 = (x − 4)(x + 2). Zeroes: 4 and −2. Sum = 4 + (−2) = 2 = −(−2)/1 = −b/a ✓ Product = 4 × (−2) = −8 = −8/1 = c/a ✓
(ii) 4s² − 4s + 1 4s² − 4s + 1 = (2s − 1)². Zeroes: 1/2 and 1/2. Sum = 1 = −(−4)/4 = −b/a ✓ Product = 1/4 = 1/4 = c/a ✓
(iii) 6x² − 3 − 7x = 6x² − 7x − 3 6x² − 7x − 3 = (3x + 1)(2x − 3). Zeroes: −1/3 and 3/2. Sum = −1/3 + 3/2 = 7/6 = −(−7)/6 = −b/a ✓ Product = (−1/3)(3/2) = −1/2 = −3/6 = c/a ✓
(iv) 4u² + 8u 4u² + 8u = 4u(u + 2). Zeroes: 0 and −2. Sum = −2 = −8/4 = −b/a ✓ Product = 0 = 0/4 = c/a ✓
(v) t² − 15 t² − 15 = (t − √15)(t + √15). Zeroes: √15 and −√15. Sum = 0 = −0/1 = −b/a ✓ Product = −15 = −15/1 = c/a ✓
(vi) 3x² − x − 4 3x² − x − 4 = (3x − 4)(x + 1). Zeroes: 4/3 and −1. Sum = 4/3 − 1 = 1/3 = −(−1)/3 = −b/a ✓ Product = (4/3)(−1) = −4/3 = c/a ✓
Q2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Use x² − (sum)x + (product).
(i) 1/4, −1 x² − (1/4)x + (−1) = x² − (1/4)x − 1 → multiply by 4: 4x² − x − 4.
(ii) √2, 1/3 x² − √2 x + 1/3 → multiply by 3: 3x² − 3√2 x + 1.
(iii) 0, √5 x² − 0·x + √5 = x² + √5.
(iv) 1, 1 x² − x + 1.
(v) −1/4, 1/4 x² − (−1/4)x + 1/4 = x² + (1/4)x + 1/4 → multiply by 4: 4x² + x + 1.
(vi) 4, 1 x² − 4x + 1.
Common Mistakes
- Forgetting the negative sign in sum of zeroes (it is −b/a, not b/a).
- Sign errors when factorising — always expand back to check.
- When forming a polynomial, writing −(product) instead of +(product) in x² − (sum)x + (product).
Exam Tips
- Always verify zeroes against α + β = −b/a and αβ = c/a — it is worth a mark and catches errors.
- For graph questions, the only rule is: count x-axis intersections.
- Quadratic factorisation (splitting the middle term) is the core skill — practise it until it is automatic.