CBSE 2025–26 | Step-by-Step Answers | Free PDF Download
Polynomials are a core concept in algebra and a high-weightage topic in Class 10 Maths board exams. This chapter helps students understand types of polynomials, zeroes, their graphical meaning, and the division algorithm. Below are complete NCERT solutions for all exercises, explained clearly for easy learning.
Exercise 2.1 – Geometrical Meaning of Zeroes
Q1. Six graphs of polynomials are provided. Students must identify the number of zeroes based on how many times the curve intersects the x-axis.
Answers:
- (i) 0 zeroes
- (ii) 1 zero
- (iii) 3 zeroes
- (iv) 2 zeroes
- (v) 4 zeroes
- (vi) 3 zeroes
📌 Concept: A zero of a polynomial is a point where the graph touches or crosses the x-axis.
Exercise 2.2 – Zeroes and Coefficients Relationship
Q1. Find the zeroes of each quadratic polynomial and verify:
- Sum of zeroes = −ba-\frac{b}{a}
- Product of zeroes = ca\frac{c}{a}
| Polynomial | Zeroes | Sum | Product |
|---|---|---|---|
| x2−2x−8x ^2 – 2x – 8 | 4, -2 | 2 | -8 |
| 4s2−4s+14s ^2 – 4s + 1 | ½, ½ | 1 | ¼ |
| 6×2−7x−36x ^2 – 7x – 3 | -⅓, ³⁄₂ | 7/6 | -½ |
| 4u2+8u4u ^2 + 8u | 0, -2 | -2 | 0 |
| t2−15t ^2 – 15 | √15, -√15 | 0 | -15 |
| 3×2−x−43x ^2 – x – 4 | 4/3, -1 | 1/3 | -4/3 |
Q2. Construct quadratic polynomials from given sum and product of zeroes.
| Sum | Product | Polynomial |
|---|---|---|
| ¼ | -1 | 4×2−x−44x^2 – x – 4 |
| √2 | ⅓ | 3×2−3√2x+13x^2 – 3√2x + 1 |
| 0 | √5 | x2−√5x^2 – √5 |
| 1 | 1 | x2−x+1x^2 – x + 1 |
| -¼ | ¼ | 4×2+x+14x^2 + x + 1 |
| 4 | 1 | x2−4x+1x^2 – 4x + 1 |
Exercise 2.3 – Division Algorithm for Polynomials
Q1. Divide p(x) by g(x) and find quotient and remainder.
| p(x) | g(x) | Quotient | Remainder |
|---|---|---|---|
| x3−3×2+5x−3x^3 – 3x^2 + 5x – 3 | x2−2x^2 – 2 | x−3x – 3 | 7x−97x – 9 |
| x4−3×2+4x+5x^4 – 3x^2 + 4x + 5 | x2+1−xx^2 + 1 – x | x2+x−3x^2 + x – 3 | 8 |
| x4−5x+6x^4 – 5x + 6 | 2−x22 – x^2 | −x2−2-x^2 – 2 | −5x+10-5x + 10 |
Q2. Check if one polynomial is a factor of another.
- t2−3t^2 – 3 is a factor of 2t4+3t3−2t2−9t−122t^4 + 3t^3 – 2t^2 – 9t – 12 ✅
- x2+3x+1x^2 + 3x + 1 is a factor of 3×4+5×3−7×2+2x+23x^4 + 5x^3 – 7x^2 + 2x + 2 ✅
- x3−3x+1x^3 – 3x + 1 is not a factor of x5−4×3+x2+3x+1x^5 – 4x^3 + x^2 + 3x + 1 ❌
Q3. Given two zeroes √(5/3) and -√(5/3), find remaining zeroes of 3×4+6×3−2×2−10x−53x^4 + 6x^3 – 2x^2 – 10x – 5
→ Remaining zeroes: -1, -1
Q4. If quotient = x−2x – 2, remainder = −2x+4-2x + 4, and dividend = x3−3×2+x+2x^3 – 3x^2 + x + 2, find g(x):
→ g(x)=x2−x+1g(x) = x^2 – x + 1
Q5. Examples satisfying division algorithm:
- (i) deg p(x) = deg q(x): p(x)=3×2+3x+3p(x) = 3x^2 + 3x + 3, g(x)=3g(x) = 3
- (ii) deg q(x) = deg r(x): p(x)=x2+3p(x) = x^2 + 3, g(x)=x−1g(x) = x – 1
- (iii) deg r(x) = 0: p(x)=x2+1p(x) = x^2 + 1, g(x)=xg(x) = x
Exercise 2.4 – Cubic Polynomials and Zeroes
Q1. Verify zeroes and their relationships:
- (i) 2×3+x2−5x+22x^3 + x^2 – 5x + 2; zeroes: ½, 1, -2
- (ii) x3−4×2+5x−2x^3 – 4x^2 + 5x – 2; zeroes: 2, 1, 1
Q2. Find cubic polynomial with:
- Sum of zeroes = 2
- Sum of product of zeroes (two at a time) = -7
- Product of zeroes = -14 → Polynomial: x3−2×2−7x+14x^3 – 2x^2 – 7x + 14
Q3. If zeroes are a−ba – b, aa, a+ba + b, and polynomial is x3−3×2+x+1x^3 – 3x^2 + x + 1, find a and b:
→ a=1a = 1, b=±√2b = ±√2
Q4. Given zeroes: 2±√32 ± √3, find remaining zeroes of x4−6×3−26×2+138x−35x^4 – 6x^3 – 26x^2 + 138x – 35
→ Remaining zeroes: -5, 7
Q5. If remainder of division is x+ax + a, and divisor is x2−2x+kx^2 – 2x + k, find k and a:
→ k=5k = 5, a=−5a = -5
❓ FAQs – Class 10 Chapter 2 Polynomials
Q1. How many exercises are there in Chapter 2 Polynomials?
There are 4 exercises: 2.1 to 2.4.
Q2. Are these solutions based on the latest CBSE syllabus?
Yes, they follow the 2025–26 NCERT guidelines.
Q3. Can I download the solutions for offline study?
Absolutely! Each exercise comes with a free PDF download.
Q4. Are these solutions enough for board exam preparation?
Yes. They’re designed to help you understand concepts deeply and solve exam-style questions confidently.
