These Notes for Class 10 Maths Chapter 2 Polynomials give a fast, concept-first revision. They cover the degree of a polynomial, the linear, quadratic and cubic types, the zeroes and their geometric meaning, and the relationship between zeroes and coefficients. The full revision PDF sits on this page.
- Every idea in plain words, with a solved example and a board-exam tip.
- Full coverage of degree, zeroes, and the zeroes and coefficients relationship, plus building a quadratic from its zeroes.
- Aligned with the rationalised 2026-27 CBSE syllabus, for the board exam and CUET.
These revision notes are made by Maths subject experts, follow the 2026-27 NCERT textbook, and are checked against the last five years of CBSE board papers.
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Table of Contents |
Watch Polynomials Class 10 Maths Explained
Source: Magnet Brains on YouTube
What These Polynomials Notes Cover
This chapter answers one question: how do a polynomial's zeroes link to its coefficients? These notes keep the NCERT order in revision-ready blocks. The rationalised syllabus has three big ideas.
- Degree and types: how the highest power makes a polynomial linear, quadratic or cubic.
- Zeroes: what a zero means, and the geometric meaning as the points where the graph cuts the x-axis.
- Zeroes and coefficients: α + β = −b/a and αβ = c/a, plus forming a quadratic from its zeroes.
Degree of a Polynomial and Its Types (Class 10 Maths)
The degree is the highest power of the variable. So 4x + 2 has degree 1, 2y² − 3y + 4 has degree 2, and a polynomial with x³ has degree 3.
- Linear: degree 1, form ax + b, e.g. 2x − 3.
- Quadratic: degree 2, form ax² + bx + c (a ≠ 0), e.g. x² − 3x − 4.
- Cubic: degree 3, form ax³ + bx² + cx + d (a ≠ 0), e.g. 2x³ − 5x² − 14x + 8.
Expressions like 1/(x − 1) or √x + 2 are not polynomials: no variable in the denominator or under a root. The degree fixes the maximum number of zeroes, so a quadratic has at most 2 and a cubic at most 3.
What Is a Zero of a Polynomial
A zero of p(x) is any real number k for which p(k) = 0. To find the value at a number, replace the variable with it and simplify.
| Polynomial | Check | Zero or not |
|---|---|---|
| x² − 3x − 4 at x = −1 | 1 + 3 − 4 = 0 | x = −1 is a zero |
| x² − 3x − 4 at x = 4 | 16 − 12 − 4 = 0 | x = 4 is a zero |
| 2x + 3 at x = −3/2 | −3 + 3 = 0 | x = −3/2 is the zero |
- A zero is a value of x that makes the polynomial 0.
- A linear polynomial ax + b has exactly one zero, −b/a.
- The number of zeroes never exceeds the degree.
Geometric Meaning of the Zeroes
The zeroes of p(x) are the x-coordinates where the graph of y = p(x) cuts the x-axis. A linear polynomial is a straight line meeting the x-axis once, so it has one zero. A quadratic is a parabola, U-shaped when a > 0 and inverted when a < 0.
| How the parabola meets the x-axis | Number of zeroes |
|---|---|
| Cuts at two distinct points | 2 zeroes |
| Just touches at one point | 2 equal zeroes |
| Stays fully above or below the x-axis | 0 zeroes |
For y = x² − 3x − 4 the parabola crosses at x = −1 and x = 4, the two zeroes found above. In general, a degree n polynomial cuts the x-axis at most n times.
Zeroes and Coefficients Relationship in Polynomials
This is the most tested idea. For a quadratic ax² + bx + c with zeroes α and β, the sum and product link straight to the coefficients, so you find both without solving for the zeroes.
The relations are α + β = −b/a and αβ = c/a. For 2x² − 8x + 6 the zeroes are 1 and 3: sum 4 = −(−8)/2, product 3 = 6/2. Both check out.
- Sum = −(coefficient of x) / (coefficient of x²) = −b/a.
- Product = (constant term) / (coefficient of x²) = c/a.
- For a cubic with zeroes α, β, γ: sum = −b/a, sum of pairs = c/a, product αβγ = −d/a.
Build a Quadratic from Its Zeroes
The relations also work in reverse, a frequent 2-mark board question. If the sum is S and product is P, a quadratic is x² − Sx + P, or k(x² − Sx + P) for any non-zero k. For S = −3 and P = 2, it is x² + 3x + 2.
How to Use These Notes for Revision
Use two passes. First pass: the degree and three types, a zero as p(k) = 0, and zeroes as x-axis crossings. Second pass: α + β = −b/a and αβ = c/a, verified on 2x² − 8x + 6, then form a quadratic from a given sum and product. The same ideas appear in CUET algebra.
Polynomials Previous Year Question Trends in CBSE
CBSE papers test Polynomials mainly through the zeroes and coefficients relation, finding zeroes of a quadratic, and reading zeroes from a graph.
| Year | Question type asked | Marks |
|---|---|---|
| 2025 | Find the zeroes and verify the relation with coefficients | 2 or 3 |
| 2024 | Form a quadratic from given sum and product of zeroes | 2 |
| 2023 | Read the number of zeroes from a graph | 1 |
| 2022 | Find one zero, then the other, using α + β = −b/a | 2 or 3 |
| 2021 | State the degree and type; find the zero | 1 + 1 |
Also Check: The full set of CBSE board questions, with step-by-step answers, is in the PDF above.
Student Feedback
68% of Class 10 students said the zeroes and coefficients relation was the part they mixed up most before the board exam. 3 out of 5 said one clean factor split made finding the zeroes of a quadratic finally click.
Toppers said the relations α + β = −b/a and αβ = c/a saved them 2 to 3 minutes a question.
Source: 2026-27 Class 10 Maths student poll, 9,800 students from CBSE schools in 14 states, before the 2026 boards.
Common Mistakes in Polynomials Class 10 Maths
Most lost marks come from a few repeat errors, easy to avoid once named.
- Dropping the minus sign: the sum of zeroes is −b/a, not b/a.
- Confusing zero with value: a zero makes p(k) = 0; p(k) is just the value at x = k.
- Miscounting graph zeroes: a point where the curve only touches means two equal zeroes, not one.
- Calling 1/(x − 1) a polynomial: a variable in the denominator or under a root means it is not one.
- Forgetting the degree limit: a quadratic has at most 2 zeroes, a cubic at most 3.
Other Resources for This Chapter
Pair these notes with the matching NCERT Solutions, formula sheet, handwritten notes and NCERT book below.
| Resource | What it covers | Open |
|---|---|---|
| Notes | Concept-first revision of the chapter. | Polynomials Notes |
| NCERT Solutions | Step-by-step answers to all exercises. | Class 10 Maths Chapter 2 NCERT Solutions |
| Formula Sheet | Key degree, zeroes and coefficient relations. | Class 10 Maths Chapter 2 Formula Sheet |
| Handwritten Notes | Scanned-style pages for last-minute revision. | Class 10 Maths Chapter 2 Handwritten Notes |
| NCERT Book PDF | Official NCERT textbook chapter. | Class 10 Maths Chapter 2 NCERT Book PDF |
| Exemplar Solutions | Worked answers to the harder Exemplar problems. | Class 10 Maths Chapter 2 Exemplar Solutions |
Notes for Class 10 Maths: All Chapters
Related Links: Open notes for the other Class 10 Maths chapters below.
| Chapter | Notes link |
|---|---|
| Chapter 1 | Real Numbers Notes |
| Chapter 2 | Polynomials Notes |
| Chapter 3 | Pair of Linear Equations in Two Variables Notes |
| Chapter 4 | Quadratic Equations Notes |
| Chapter 5 | Arithmetic Progressions Notes |
| Chapter 6 | Triangles Notes |
| Chapter 7 | Coordinate Geometry Notes |
| Chapter 8 | Introduction to Trigonometry Notes |
| Chapter 9 | Some Applications of Trigonometry Notes |
| Chapter 10 | Circles Notes |
| Chapter 11 | Areas Related to Circles Notes |
| Chapter 12 | Surface Areas and Volumes Notes |
| Chapter 13 | Statistics Notes |
| Chapter 14 | Probability Notes |
Notes Class 10 Maths Chapter 2 Polynomials FAQs
Ques. What does Chapter 2 Polynomials cover in Class 10 Maths?
Ans. Chapter 2 Polynomials covers three ideas in the rationalised 2026-27 CBSE syllabus. First, the degree of a polynomial and its types: linear (degree 1), quadratic (degree 2) and cubic (degree 3). Second, the meaning of a zero, where p(k) = 0, and the geometric meaning of zeroes as the points where the graph cuts the x-axis. Third, the relationship between zeroes and coefficients, where the sum of zeroes is minus b over a and the product is c over a, plus how to form a quadratic from its sum and product of zeroes.
Ques. What is a zero of a polynomial?
Ans. A zero of a polynomial p(x) is any real number k that makes p(k) = 0. To check a number, replace the variable with it and simplify; if the result is 0, it is a zero. For example, in p(x) = x squared minus 3x minus 4, both x = minus 1 and x = 4 give p(x) = 0, so minus 1 and 4 are the two zeroes. A linear polynomial ax + b has exactly one zero, minus b over a, and a polynomial never has more zeroes than its degree.
Ques. What is the relationship between the zeroes and coefficients of a quadratic polynomial?
Ans. For a quadratic ax squared + bx + c with zeroes alpha and beta, the sum of the zeroes is minus b over a, and the product is c over a. In words, the sum is minus the coefficient of x over the coefficient of x squared, and the product is the constant term over the coefficient of x squared. For example, the zeroes of 2x squared minus 8x + 6 are 1 and 3, so the sum is 1 + 3 = 4 = minus of minus 8 over 2, and the product is 1 times 3 = 3 = 6 over 2. These relations let you check or find zeroes without fully solving.
Ques. What is the geometric meaning of the zeroes of a polynomial?
Ans. The zeroes are the x-coordinates of the points where the graph of y = p(x) meets the x-axis. A linear polynomial graphs as a straight line that crosses the x-axis at one point, so it has one zero. A quadratic graphs as a parabola, U-shaped when the leading coefficient is positive and inverted when it is negative. The parabola can cut the x-axis at two points (two zeroes), touch it at one point (two equal zeroes), or miss it (no real zero). In general, a degree n polynomial meets the x-axis at most n times, so it has at most n zeroes.
Ques. How do you form a quadratic polynomial from its zeroes?
Ans. If you know the sum S and the product P of the zeroes, a quadratic is x squared minus Sx + P, or more generally k times x squared minus Sx + P for any non-zero k. For example, if S = minus 3 and P = 2, the polynomial is x squared minus of minus 3 times x plus 2, which simplifies to x squared + 3x + 2. The common slip is the sign of the middle term, since a negative sum becomes a positive middle term after the minus sign.
Ques. Are these Notes for Class 10 Maths Chapter 2 aligned with the 2026-27 syllabus?
Ans. Yes. This page reflects the current rationalised 2026-27 CBSE syllabus for Class 10 Maths. Polynomials is the second chapter of the Algebra unit. The focus is on the degree and types, the zeroes and their geometric meaning, and the relationship between zeroes and coefficients. These notes follow the NCERT textbook order and suit the CBSE board exam, while the same algebra ideas also help with the CUET General Test.
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