Chapter 2: Polynomials 10th Mathematics

 

Real Numbers

Mathematics – Class 10
Reprint Edition: 2025–26

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Chapter 2: Polynomials

2.1 Introduction

In earlier classes, you have studied expressions like $x + 1$, $x^2 + 2x + 1$, etc. These are known as algebraic expressions. In this chapter, we will focus on a special type of algebraic expressions called polynomials.

You will learn:

• What polynomials are

• Different types of polynomials based on degree

• Operations on polynomials

• Division algorithm for polynomials

• Finding zeros of polynomials

• Relationship between zeros and coefficients

Let us begin with understanding polynomials.

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2.2 Polynomials in One Variable

A polynomial in one variable $x$ is an algebraic expression of the form:

$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

where:

• $a_n, a_{n-1}, \ldots, a_1, a_0$ are real numbers (called coefficients),

• $a_n \ne 0$, and

• $n$ is a non-negative integer (called the degree of the polynomial).

Example:

• $2x + 3$ is a polynomial of degree 1

• $4x^2 - 5x + 6$ is a polynomial of degree 2

• $x^3 + 2x^2 - x + 7$ is a polynomial of degree 3

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2.3 Zeros of a Polynomial

A zero of a polynomial $p(x)$ is a number $a$ such that $p(a) = 0$.

Example:

Let $p(x) = x^2 - 4$. Then, $p(2) = 2^2 - 4 = 0 \quad \text{and} \quad p(-2) = (-2)^2 - 4 = 0$ So, 2 and -2 are zeros of the polynomial.

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2.4 Remainder Theorem

If $p(x)$ is a polynomial of degree greater than or equal to one and is divided by $x - a$, then the remainder is $p(a)$.

Example:

Let $p(x) = x^2 - 3x + 2$. Divide by $x - 1$. Then the remainder is $p(1) = 1^2 - 3(1) + 2 = 0$

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2.5 Factor Theorem

If $p(a) = 0$, then $x - a$ is a factor of $p(x)$, and vice versa.

Example:

$p(x) = x^2 - 5x + 6$ Check $p(2) = 0$ and $p(3) = 0$ $\Rightarrow (x - 2)(x - 3)$ is the factorised form of $p(x)$

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2.6 Relationship Between Zeros and Coefficients

For a quadratic polynomial $ax^2 + bx + c$, if $\alpha$ and $\beta$ are its zeros, then:

• Sum of the zeros: $\alpha + \beta = -\frac{b}{a}$

• Product of the zeros: $\alpha\beta = \frac{c}{a}$

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2.7 Division Algorithm for Polynomials

If $p(x)$ and $g(x)$ are polynomials such that $g(x) \ne 0$, then there exist polynomials $q(x)$ and $r(x)$ such that:

$p(x) = g(x)q(x) + r(x)$

where the degree of $r(x)$ is less than the degree of $g(x)$.

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End of Chapter 2