---
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id: "69f86844293c3123114d4de1"
title: "Polynomials"
board: "CBSE"
curriculum: "CBSE"
class: "Class 10"
subject: "Mathematics"
book: "Mathematics"
chapter: "Polynomials"
chapter_slug: "polynomials"
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last_updated: "2026-06-20"
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# Polynomials
In Class IX, students studied polynomials in one variable and their degrees. A polynomial in $x$ is defined as an expression where the highest power of $x$ determines its degree.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Polynomials |
| Pages | 9-23 |

---

## Chapter Summary

### Short Summary
This chapter explores the definition and types of polynomials, including linear, quadratic, and cubic polynomials, as well as the concept of zeroes and their connection to polynomial coefficients.

### Detailed Summary
The chapter provides a detailed look into polynomials, emphasizing their degrees and types. A polynomial of degree 1 is classified as a linear polynomial, degree 2 as a quadratic polynomial, and degree 3 as a cubic polynomial. The relation of zeroes to polynomial coefficients is examined, exemplified by $p(x) = x^2 - 3x - 4$, where the zeroes -1 and 4 indicate distinct values of the polynomial. Additionally, the chapter introduces the division algorithm for polynomials.

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## Topic-Wise Explanation

### Introduction to Polynomials
Polynomials are algebraic expressions consisting of variables raised to non-negative integer powers.

### Geometrical Meaning of the Zeroes of a Polynomial
Zeroes represent the x-values where the polynomial evaluates to zero, delving into their graphical representation.

### Linear Polynomial Zeroes
For linear polynomials, the zero can be found using $k = -b/a$ where $p(x) = ax + b$.

### Quadratic Polynomial Zeroes
Zeroes of quadratic polynomials relate directly to their coefficients, explored through examples like $p(x) = x^2 - 3x - 4$.

### Relationship between Zeroes and Coefficients
This section establishes the mathematical correlation between the zeroes of polynomials and their coefficients, confirmed through polynomial forms.

### Quadratic Polynomial Coefficient Relations
Details the relationships for quadratic coefficients with examples.

### Cubic Polynomial Coefficient Relations
Discusses cubic polynomial coefficients and their significance in determining polynomial behavior.

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## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Degree Definition | The degree of a polynomial is the highest power of its variable. |
| Linear Polynomial | A polynomial with degree 1, expressed as $ax + b$. |
| Quadratic Polynomial | A polynomial with degree 2, typically structured as $ax^2 + bx + c$. |
| Cubic Polynomial | An expression of degree 3, formatted as $ax^3 + bx^2 + cx + d$. |
| Zeroes Definition | Zeroes are the solutions $k$ for which $p(k) = 0$. |

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## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Polynomial | An algebraic expression involving a sum of powers in one or more variables. |
| Zeroes | The points at which a polynomial evaluates to zero. |
| Coefficients | The numerical factors in the terms of a polynomial. |

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## Important Points for Revision

* Polynomials are expressions with non-negative integer powers.
* The degree of a polynomial determines its classification.
* Linear polynomials are defined as having a degree of 1.
* Quadratic polynomials are defined as having a degree of 2.
* The general form of a quadratic polynomial is $ax^2 + bx + c$.
* Cubic polynomials have a degree of 3, represented as $ax^3 + bx^2 + cx + d$.
* Zeroes are found by solving the equation $p(x) = 0$.
* The relationship between zeroes and coefficients establishes significant connections in polynomial calculus.
* The division algorithm allows for polynomial division, enhancing understanding of polynomial behavior.

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## Practice Questions

### Short Answer Questions
1. Define a polynomial.
2. What is the degree of the polynomial $5x^4 - 3x + 7$?
3. Identify the zero of the linear polynomial $2x + 3$.
4. What is a quadratic polynomial?
5. How do you find the zeroes of a cubic polynomial?

### Long Answer Questions
1. Explain the relationship between the zeroes and coefficients of a quadratic polynomial with examples.
2. Demonstrate the use of the division algorithm on a polynomial expression.
3. Discuss the implications of the degree of a polynomial on its graph and zeroes.

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## Related Concepts

* Linear Polynomial
* Quadratic Polynomial
* Cubic Polynomial
* Division Algorithm

---

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 69f86844293c3123114d4de1 |
| Canonical URL | https://www.edzy.ai/cbse-class-10-mathematics-polynomials |
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