Chapter 2: Introduction to Linear Polynomials

Chapter 2: Introduction to Linear Polynomials

Class 9 Mathematics Notes

2.1 Introduction

In earlier classes, we studied algebraic expressions. In this chapter, we study a special type of algebraic expression called a linear polynomial.

Algebraic Expression

An algebraic expression is a combination of:

• numbers,

• variables,

• constants,

• coefficients,

• mathematical operations.

Example

If Raju buys x red boxes and each red box contains 4 pens, then total pens are: 4x

If he buys y blue boxes and each blue box contains 5 pencils, then total pencils are: 5y

If he also gets 3 extra pens, then total items are: 4x + 5y + 3

Here:

Terms, Variables, Coefficients and Constants

1. Term

Each part of an algebraic expression separated by + or − sign is called a term.

Example: In 4x + 5y + 3, the terms are: 4x, 5y and 3

2. Variable

A symbol or letter whose value can change is called a variable.

Examples: x, y, z, a, b

3. Coefficient

The numerical factor multiplied with a variable is called its coefficient.

Example: In 7x, the coefficient of x is 7.

In −5y, the coefficient of y is −5.

4. Constant

A number that has a fixed value is called a constant.

Example: In 3x + 8, the constant term is 8.

One-variable Polynomial

An algebraic expression involving only one variable and its powers is called a one-variable polynomial or univariate polynomial.

Examples: 4x, x² + 1, 2y − 5, 5y³ + y² + 2y − 1

Degree of a Polynomial

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

Examples

2.2 Linear Polynomials

A polynomial of degree 1 is called a linear polynomial.

General form

A linear polynomial is generally written as: ax + b

where:

• x is the variable,

• a and b are constants,

• a ≠ 0.

Examples of Linear Polynomials

2x + 3, 5x − 7, 4y, 9 − 3z

Non-examples

Linear Polynomial as an Input-Output Process

A polynomial can be understood as an input-output machine.

Example: For the linear polynomial: 2x + 3

If input x = 4, then: 2 × 4 + 3 = 11

So, output is 11.

If input x = −6, then: 2 × (−6) + 3 = −12 + 3 = −9

So, output is −9.

Linear Polynomial and Linear Equation

When we equate a linear polynomial to a constant, we get a linear equation.

Example: 2x + 10 = 64

Here, 2x + 10 is a linear polynomial.

Solving: 2x + 10 = 64
2x = 54
x = 27

2.3 Exploring Linear Patterns

A linear pattern is a sequence of numbers in which the difference between two consecutive terms is constant.

Example: Pattern of square tiles

Stages:

The sequence is:

1, 3, 5, 7, 9, 11, 13, ...

Here, each term increases by 2.

So, it is a linear pattern.

Rule for nth stage

Number of tiles at stage n:

2n − 1

For example:

15th stage:

2 × 15 − 1 = 30 − 1 = 29

26th stage:

2 × 26 − 1 = 52 − 1 = 51

Linear Pattern in Daily Life

Example: Pocket money spending

Bela has ₹100 and spends ₹5 every day.

Amount left after n days: 100 − 5n

This is a linear polynomial because the amount decreases by a fixed amount, ₹5, every day.

2.4 Linear Growth and Linear Decay

Linear expressions help us understand situations where a quantity increases or decreases by a fixed amount.

Linear Growth

A pattern in which a quantity increases by a fixed amount over equal intervals is called linear growth.

Example Cost of a journey is given by: C(d) = 100 + 60d

where:

• C is total cost,

• d is distance travelled in km.

Here, cost increases by ₹60 for every 1 km.

So, this is linear growth.

Linear Decay

A pattern in which a quantity decreases by a fixed amount over equal intervals is called linear decay.

Example Height of water in a tank is given by: h(t) = 3 − 0.5t

where:

• h is height of water,

• t is time in months.

Here, height decreases by 0.5 m every month.

So, this is linear decay.

Difference between Linear Growth and Linear Decay

2.5 Linear Relationships

A linear relationship represents the relationship between two variables.

It is written in the form: y = ax + b

where:

• x and y are variables,

• a is the slope,

• b is the y-intercept.

Example: Internet Bill

A telecom company charges a fixed monthly fee and extra cost per GB of data.

Given:

• When data used is 10 GB, bill is ₹350.

• When data used is 20 GB, bill is ₹550.

Let the relationship be: y = ax + b

where:

• x = data used in GB,

• y = monthly bill.

Using the given values:

350 = 10a + b

550 = 20a + b

Subtracting:

550 − 350 = 20a − 10a

200 = 10a

a = 20

Now, 350 = 10 × 20 + b

350 = 200 + b

b = 150

So, the linear relationship is:

y = 20x + 150

Meaning:

• ₹20 is charged per GB.

• ₹150 is the fixed monthly fee.

2.6 Visualising Linear Relationships

A linear relationship can be shown on a graph.

The graph of a linear equation is always a straight line.

General form

y = ax + b

To draw the graph of a line, we need at least two points.

Example: Graph of y = 2x + 1

Choose two values of x.

Plot the points (0, 1) and (3, 7) on the coordinate plane.

Join them and extend the line in both directions.

This gives the graph of:

y = 2x + 1

Points on a Line

A point lies on a line if its coordinates satisfy the equation of the line.

Example:

Check whether (7, 15) lies on the line:

y = 2x + 1

Substitute x = 7:

y = 2 × 7 + 1 = 14 + 1 = 15

Since y = 15, the point (7, 15) lies on the line.

Slope of a Line

In the equation:

y = ax + b

a is called the slope of the line.

The slope tells us how steep the line is.

Important points

y-intercept

In the equation:

y = ax + b

b is called the y-intercept.

The y-intercept is the point where the line cuts the y-axis.

Since every point on the y-axis has x-coordinate 0, the line cuts the y-axis at:

(0, b)

Examples

Effect of Changing a and b

In the equation: y = ax + b

1. Changing a

If a changes, the slope changes.

This means the steepness of the line changes.

Example:

y = x, y = 2x, y = 3x

All pass through the origin, but their steepness is different.

2. Changing b

If b changes but a remains the same, the lines remain parallel.

Example:

y = 2x − 1, y = 2x + 1, y = 2x + 5

All have the same slope 2, so they are parallel.

Important Formulae and Results

Key Differences

Linear Polynomial vs Quadratic Polynomial

Linear Growth vs Linear Decay

Solved Examples

Example 1: Find degree

Find the degree of: 4x − 3

The highest power of x is 1.

Answer: Degree = 1, so it is a linear polynomial.

Example 2: Evaluate linear polynomial

Find the value of: 5x − 3, when x = 2

Substitute x = 2: 5 × 2 − 3 = 10 − 3 = 7

Answer: 7

Example 3: Linear growth

A plant is 1.75 feet tall and grows 0.5 feet every month.

Height after t months: h = 1.75 + 0.5t

After 7 months: h = 1.75 + 0.5 × 7

h = 1.75 + 3.5 = 5.25 feet

Answer: 5.25 feet

Example 4: Linear decay

A phone costs ₹10,000. Its value decreases by ₹800 every year.

Value after t years: v = 10000 − 800t

After 3 years: v = 10000 − 800 × 3

v = 10000 − 2400 = 7600

Answer: ₹7600

Chapter Summary

1. An algebraic expression is made using variables, constants, coefficients and operations.

2. A polynomial in one variable is called a univariate polynomial.

3. The highest power of the variable in a polynomial is called its degree.

4. A polynomial of degree 1 is called a linear polynomial.

5. The general form of a linear polynomial is ax + b, where a ≠ 0.

6. A linear pattern is a sequence in which the difference between consecutive terms is constant.

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

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

9. A linear relationship between two variables is written as y = ax + b.

10. The graph of a linear relationship is always a straight line.

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

12. Lines with the same slope but different y-intercepts are parallel.

13. Linear growth is shown by a straight line with positive slope.

14. Linear decay is shown by a straight line with negative slope.