Chapter Notes: Polynomials

# Polynomials Class 9 Notes Maths Chapter 2

 Table of contents Introduction Polynomials in One Variable Zeroes of a Polynomial Factorization of Polynomials Algebraic Identities

## Introduction

In mathematics, a variable is denoted by a symbol that can take any real value, often represented by letters such as x, y, z, etc. Expressions like 2x, 3x, -x, and -1/2x are examples of algebraic expressions, specifically in the form (a constant) × x. When the constant is unknown, it is denoted as a, b, c, etc.

A polynomial is an algebraic expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication operations.

• Example: 2�3−5�2+3�−72x35x2+3x7 is a polynomial.

“The expression which contains one or more terms with non – zero coefficient is called polynomial”

### Understanding Expressions with Examples

• Consider a square of side 3 units, where the perimeter is given by the sum of the lengths of its four sides.
• If each side is x units, the perimeter is expressed as 4x units, showcasing how the value of the variable influences the result.
• The area of the square, denoted as x² square units, is an example of an algebraic expression.

## Polynomials in One Variable

Polynomials are algebraic expressions with variables, coefficients, and exponents. When the exponents are whole numbers, the expressions are termed polynomials in one variable.

Example: x³ - x² + 4x + 7 and 3y² + 5y.

Terms and Coefficients

In a polynomial like x² + 2x, x² and 2x are referred to as terms. Each term has a coefficient—in -x³ + 4x² + 7x - 2, coefficients are -1, 4, 7, and -2. The term x⁰ (where x⁰ = 1) is also present.

Constant Polynomials and Zero Polynomials

Constants like 2, -5, and 7 are examples of constant polynomials. The constant polynomial 0 is termed the zero polynomial, a significant concept in polynomial theory.

### Non-Polynomial Expressions

Expressions like x + 2/x, x⁻¹, and x + 3√(x) aren't polynomials due to non-whole number exponents.

### Notation for Polynomials

Polynomials can be denoted by symbols like p(x), q(x), or r(x), where the variable is x. Examples include:

1. �(�)=2�2+5�−3p(x)=2x2+5x3
2. �(�)=�3−1q(x)=x31
3. �(�)=�3+�+1r(y)=y3+y+1
4. �(�)=2−�−�2+6�5s(u)=2uu2+6u5

### Degree of a Polynomial

The degree of a polynomial is the highest power of its variable. For example, in 3x⁷ - 4x⁶ + x + 9, the degree is 7. Constant polynomials have a degree of 0.

Examples:

�(�)=�5−�4+3

Question for Chapter Notes: Polynomials
Try yourself:
What is a polynomial?

### Linear, Quadratic, and Cubic Polynomials

• Linear Polynomial: Degree 1, written as ��+�ax+b where �≠0a is not equal to 0. Examples: 2�−12x1, 2�+12y+1, 2−�2u
• Quadratic Polynomial: Degree 2, expressed as ��2+��+�ax2+bx+c where �≠0a is not equal to 0. Examples: 5−�25y2, 4�+5�24y+5y2, 6−�−�26yy2
• Cubic Polynomial: Degree 3, in the form ��3+��2+��+�ax3+bx2+cx+d where �≠0a is not equal to zero 0. Examples: 4�34x3, 2�3+12x3+1, 5�3+�25x3+x2

### General Form of a Polynomial

A polynomial in one variable of degree n is written as: ����+��−1��−1+…+�1�+�0anxn+an1xn-1++a1x+a0 where �0,�1,…,��a0,a1,an are constants and ��≠0an=0.

### Zero Polynomial and Beyond

The zero polynomial, denoted as 0, has an undefined degree. Polynomials can extend to more than one variable, like �2+�2+���x2+y2+xyz in three variables.

## Zeroes of a Polynomial

Consider the polynomial �(�)=5�3−2�2+3�−2p(x)=5x32x2+3x2. To find the value of �(�)p(x) at different points, substitute the given values for x

For Example, �(1)

Given Polynomial: p(x) = 5x3 - 2x2 + 3x - 2

Example Calculation:

1. For x = 1:

p(1) = 5 - 2 + 3 - 2 = 4

2. For x = 0:

p(0) = 0 - 0 + 0 - 2 = -2

3. For x = -1:

p(-1) = -5 - 2 - 3 - 2 = -12

In summary, for the given polynomial �(�)=5�3−2�2+3�−2p(x)=5x32x2+3x2:

• �(1)=4p(1)=4
• �(0)=−2p(0)=2
• �(−1)=−12p(1)=12

These values are found by substituting the respective values of x into the polynomial expression.

Example: Value of Polynomials at Given Points

(i) For �(�)=5�2−3�+7p(x)=5x23x+7 at �=1x=1: �(1)=5(1)2−3(1)+7=9p(1)=5(1)23(1)+7=9

(ii) For �(�)=3�3−4�+11q(y)=3y34y+11 at �=2y=2: �(2)=3(2)3−4(2)+11=27−8+11=30q(2)=3(2)34(2)+11=278+11=30

(iii) For �(�)=4�4+5�3−�2+6p(t)=4t4+5t3t2+6 at �=�t=a: �(�)=4�4+5�3−�2+6p(a)=4a4+5a3a2+6

### Identifying Zeros of Polynomials

When evaluating �(�)=�−1p(x