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Introduction to Linear Polynomials

Class 9 Mathematics • Ganita Manjari • Chapter 2

Chapter Overview

In this chapter, we learn about polynomials, their terms, coefficients, constants and degrees. We also learn about linear polynomials and how algebraic expressions can be used to describe mathematical patterns and relationships.

You will learn how to identify polynomials, find their degree and coefficients, evaluate expressions, understand linear patterns and study linear relationships.

📘 Complete Notes

Chapter concepts, explanations, examples and revision notes.

❓ Important Questions

Important practice questions with answers and Hindi explanations for examination preparation.

📝 Exercises

Exercise 2.1 से End-of-Chapter तक सभी textbook exercises यहाँ मिलेंगी।

Topics Covered

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Quick Revision

1. Polynomials

A polynomial is an algebraic expression made up of one or more terms.

3x² + 5x - 7
Easy Idea:

A polynomial is made up of terms that are added or subtracted.

2. Terms of a Polynomial

The separate parts of a polynomial are called its terms.

4x³ - 7x² + 5x - 9

Its terms are:

3. Coefficients

The numerical factor multiplying a variable is called its coefficient.

Example

6x²

Coefficient of x² = 6

-4x

Coefficient of x = -4

4. Constant Term

A term without a variable is called the constant term.

5x³ + 2x² - 8x + 6

Constant term = 6

5. Degree of a Polynomial

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

Example

5x³ + 2x² - 4x + 8

Highest power = 3

Therefore, degree = 3

Remember:

A non-zero constant polynomial has degree 0.

6. Linear Polynomial

A polynomial of degree 1 is called a linear polynomial.

ax + b

Examples

3x + 5
-2x + 7
9x - 4

7. Value of a Polynomial

To find the value of a polynomial, substitute the given value of the variable.

Example

p(x) = 3x + 2

For x = 4:

p(4) = 3(4) + 2
p(4) = 14

8. Linear Patterns

A pattern is linear when the difference between consecutive values remains constant.

Example

5, 8, 11, 14, 17, ...

Each term increases by 3.

Therefore, it is a linear pattern.

9. Linear Growth

A quantity shows linear growth when it increases by a fixed amount in equal intervals.

Example

A plant is 20 cm tall and grows 5 cm each month.

Height = 20 + 5n

10. Linear Decay

A quantity shows linear decay when it decreases by a fixed amount in equal intervals.

Example

Value = Initial Value - 500n

11. Linear Relationships

A linear relationship describes a constant rate of change between two quantities.

Easy Idea:

If one quantity changes by a fixed amount whenever the other quantity changes by one unit, the relationship can be linear.

12. The Form y = ax + b

A linear relationship can often be written in the form:

y = ax + b

Example

y = 3x + 2

Here a = 3 and b = 2.

13. Graphs of Linear Relationships

A linear relationship can be represented by a straight line on a graph.

Example

y = 2x + 1

Choose values of x, calculate y and plot the resulting points.

Graph Tip:

Always check your coordinates before drawing the line.

14. Real-Life Applications

Linear relationships and polynomials can be useful in many practical situations.

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