CBSE Class 9 Maths Revision Notes Chapter 2: Introduction to Linear Polynomials

A linear polynomial is a polynomial of degree 1, usually written in the form ax + b.
In Class 9 Maths Chapter 2, students learn algebraic expressions, linear patterns, growth, decay, relationships and straight-line graphs.

CBSE Class 9 Maths Revision Notes Chapter 2 help students revise Introduction to Linear Polynomials from the new NCERT Ganita Manjari textbook for 2026-27. This chapter explains algebraic expressions, variables, coefficients, constants, univariate polynomials, degree of a polynomial, linear polynomials, linear equations, linear patterns, linear growth, linear decay, linear relationships, slope, y-intercept and graphs of straight lines. NCERT states that Chapter 2 develops understanding of algebraic expressions and linear polynomials through patterns, linear relationships and graphical representations.

Key Takeaways

• Algebraic Expression: An algebraic expression combines numbers, variables and operation symbols.
• Variable: A variable is a letter that represents a changing or unknown value.
• Coefficient: A coefficient is the numerical factor of a variable term.
• Constant: A constant is a fixed number in an algebraic expression.
• Univariate Polynomial: A polynomial in one variable is called a univariate polynomial.
• Degree: The highest power of the variable in a polynomial is called its degree.
• Linear Polynomial: A polynomial of degree 1 is called a linear polynomial.
• Linear Pattern: A linear pattern has a constant difference between consecutive terms.
• Linear Growth: A quantity shows linear growth when it increases by a fixed amount over equal intervals.
• Linear Decay: A quantity shows linear decay when it decreases by a fixed amount over equal intervals.
• Linear Relationship: A linear relationship between x and y is written as y = ax + b.
• Slope: In y = ax + b, a represents the slope of the line.
• y-intercept: In y = ax + b, b represents the point where the line cuts the y-axis.
• Graph: A linear equation in two variables forms a straight line.

CBSE Class 9 Maths Revision Notes Chapter 2 Structure 2026-27

Class 9 Maths Revision Notes Chapter 2: Chapter Overview

Chapter 2 of Ganita Manjari is titled Introduction to Linear Polynomials. It starts with familiar algebraic expressions and then focuses on one-variable polynomials.

The chapter uses examples from boxes of pens and pencils, garden fencing, square perimeters, chess club fees, tile patterns, pocket money, auto-rickshaw fare, mobile recharge, temperature conversion and graphs. These examples show how linear polynomials model daily-life situations.

Searches like Class 9 Maths Chapter 2 Linear Polynomials Notes, Ganita Manjari Chapter 2 Introduction to Linear Polynomials, and Class 9 Maths new NCERT book 2026 notes usually need concise definitions, formulas, tables and graph rules.

Algebraic Expressions Class 9

An algebraic expression combines numbers, variables and operation symbols. For example, 4x + 5y + 3 is an algebraic expression.

In the expression 4x + 5y + 3, the terms are 4x, 5y and 3. The variables are x and y. The coefficients of x and y are 4 and 5, while 3 is the constant term.

Terms, Variables and Coefficients Class 9

Quick Revision Note

Variables were called letter-numbers in earlier grades. In Class 9, the chapter uses the standard word variables.

Polynomials Class 9 Notes

A polynomial is an algebraic expression in one variable and its powers. The powers of the variable must be non-negative whole numbers.

NCERT uses the terms one-variable polynomial, univariate polynomial and polynomial when the context is clear. The word univariate means having one variable.

Examples of Polynomials

Degree of Polynomial Class 9

The degree of a polynomial is the highest power of the variable in that polynomial.

For example, 5y³ + y² + 2y - 1 has degree 3 because the highest power of y is 3. The polynomial x² + 5x + 1 has degree 2.

Types of Polynomials by Degree

Revision Tip

To find the degree, look only at the highest power of the variable. Do not add coefficients or terms.

Linear Polynomial Class 9

A linear polynomial is a polynomial of degree 1. It is usually written in the form ax + b, where a ≠ 0.

Examples of linear polynomials include 3z + 7, 2x + 3, 5 - 4y and 200 + 50m. NCERT states that polynomials of degree 1 are called linear polynomials.

Linear Polynomial Examples

Non-Examples

Linear Equation Class 9

When a linear polynomial in one variable is equated to a constant, it becomes a linear equation.

For example, the polynomial 2x + 10 becomes a linear equation when written as 2x + 10 = 64. NCERT uses this form to solve the example of two numbers whose sum is 64.

Example

If the smaller number is x, and the larger number is x + 10, then:

x + (x + 10) = 64

2x + 10 = 64

2x = 54

x = 27

The numbers are 27 and 37.

Linear Polynomials as Input-Output Processes

A polynomial can work like an input-output machine. The input is the value of the variable, and the output is the value of the polynomial.

For the linear polynomial 2x + 3, if x = 4, the output is 2 × 4 + 3 = 11. If x = -6, the output is 2 × (-6) + 3 = -9.

Input-Output Table for 2x + 3

Revision Tip

Substitute the value of the variable carefully. Negative values need extra attention.

Linear Pattern Class 9 Maths

A linear pattern is a sequence where the difference between consecutive terms is constant.

In the tile pattern from NCERT, the number of square tiles is 1, 3, 5, 7, 9, 11, 13. Each stage adds 2 more tiles. The expression for Stage n is 2n - 1.

Tile Pattern Table

Quick Revision Note

The expression 2n - 1 is a linear polynomial because its degree is 1.

Linear Growth and Linear Decay Class 9

Linear growth means a quantity increases by a fixed amount over equal intervals. Linear decay means a quantity decreases by a fixed amount over equal intervals.

NCERT explains linear growth using cost of journey and linear decay using water height in a tank. In both cases, the change over equal intervals is constant.

Difference Between Linear Growth and Linear Decay

Linear Growth Example

C(d) = 100 + 60d

Here, the cost increases by ₹60 for every additional kilometre.

Linear Decay Example

h(t) = 3 - 0.5t

Here, the water height decreases by 0.5 m every month.

Linear Relationship Class 9

A linear relationship between two variables x and y can be written as y = ax + b.

The expression shows how one quantity changes with another quantity. In the tile pattern, if x represents the stage number and y represents the number of square tiles, the relation is y = 2x - 1.

General Form

y = ax + b

Example from Internet Bill

If a monthly bill depends on data used, and the relation is y = 20x + 150, then:

This follows NCERT’s telecom bill example.

Graph of Linear Equation Class 9

The graph of a linear equation is a straight line. To draw it, students need any two points on the line.

For example, to draw y = 2x + 1, take x = 0, then y = 1, giving point (0, 1). Take x = 3, then y = 7, giving point (3, 7). Join these two points and extend the line.

Steps to Draw a Linear Graph

1. Choose two convenient values of x.
2. Substitute each value in the equation.
3. Find the corresponding y values.
4. Write the points as ordered pairs.
5. Plot the two points on graph paper.
6. Join the points with a straight line.
7. Extend the line in both directions.

Example Table for y = 2x + 1

Slope and y-intercept Class 9

In the equation y = ax + b, a is the slope and b is the y-intercept.

The slope tells how steep the line is. The y-intercept tells where the line cuts the y-axis. NCERT states that any line written as y = ax + b cuts the y-axis at (0, b).

Slope and y-intercept Table

Important Graph Rules

Parallel Lines Class 9

Parallel lines in this chapter have the same slope but different y-intercepts.

NCERT explains that lines of the form y = ax + b remain parallel when a is fixed and b changes. For example, y = 2x - 1, y = 2x + 1 and y = 2x + 5 are parallel lines.

Examples of Parallel Lines

Class 9 Maths Chapter 2 Formulas

Class 9 Maths Chapter 2 formulas are based on polynomials, linear relationships, graphs, slope and y-intercept.

Class 9 Maths Linear Polynomials Notes: Common Mistakes

Linear polynomials become easier when students identify the degree, slope and y-intercept correctly. Most mistakes happen while reading signs or confusing expressions with equations.

Class 9 Maths Chapter 2 Short Notes

Use these short notes for quick revision before school tests.

1. An algebraic expression combines numbers, variables and operations.
2. A variable is a letter that represents a value.
3. A coefficient is the numerical factor of a variable term.
4. A constant has no variable.
5. A univariate polynomial has one variable.
6. The degree of a polynomial is the highest power of the variable.
7. A degree 1 polynomial is called a linear polynomial.
8. A linear polynomial is written as ax + b, where a ≠ 0.
9. A linear pattern has a constant difference between consecutive terms.
10. Linear growth shows fixed increase over equal intervals.
11. Linear decay shows fixed decrease over equal intervals.
12. A linear relationship is written as y = ax + b.
13. In y = ax + b, a is the slope.
14. In y = ax + b, b is the y-intercept.
15. The graph of a linear equation is a straight line.
16. Lines with the same slope and different y-intercepts are parallel.

Chapter Summary: Introduction to Linear Polynomials

The NCERT chapter summary states that algebraic expressions combine numbers, variables and operation symbols. It also states that univariate polynomials are algebraic expressions in one variable, and the highest power of the variable gives the degree.

NCERT Class 9 Maths Ganita Manjari 2026 Chapter Solutions