Class 9 Maths Chapter 2: Introduction to Linear Polynomial Notes

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Why This Chapter Confuses Class 9 Students

The new Ganita Manjari Class 9 Chapter 2 is titled Introduction to Linear Polynomial, and it is quite different from the polynomial content that used to appear in Class 9 last year. Students clicking on old YouTube videos end up watching outdated content, which adds to the confusion about what to study and what to skip. This article follows the actual topics inside the new chapter: algebraic expressions, linear polynomials, linear equations, patterns, linear growth and decay, and visualising linear relationships.

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables (letters like x, y, z), and operation signs (+, −, ×). Each individual part of an algebraic expression — such as 4x, 5y, or 3 in the expression 4x + 5y + 3 — is called a term.

Terms are of two kinds:

• Variable term — a term attached to a letter, such as 4x or 5y
• Constant term — a fixed number with no variable attached, such as 3

In 4x + 5y + 3, the number 4 is the coefficient of x, and 5 is the coefficient of y. The variables here are x and y, while 3 is the constant.

Example: Pens and Pencils

If a shop has x boxes of red pens with 4 pens per box, y boxes of pencils with 5 pencils per box, and 3 extra free pens, the total items can be written as the algebraic expression 4x + 5y + 3.

Example: Cost of Fencing a Garden

A rectangular garden has length L metres and width W metres. It needs to be fenced and decorated:

Since there are two lengths and two widths to fence:

• Cost of wire fencing = 2L × 100 = 200L
• Cost of wooden fencing = 2W × 80 = 160W
• Cost of seeding the area = L × W × 50 = 50LW

Total cost = 200L + 160W + 50LW

Example: Wire Bent Into a Rectangle

A wire is bent to form a rectangle. If the two sides add up to 10 cm (for example, 7 cm and 3 cm, or 5.5 cm and 4.5 cm), and the length is taken as x cm, then the width becomes (10 − x) cm.

Area = x(10 − x) = 10x − x² cm²

Always include the unit — leaving it out costs marks in exams.

What Is a Polynomial?

A polynomial is an expression with one or more terms, where the coefficients are non-zero real numbers and the exponent of the variable is always a whole number (0, 1, 2, 3, …). The exponent can never be negative or a fraction.

For example, in the general form ax^n + bx^(n−1) + cx^(n−2) + …, the coefficients a, b, c must be non-zero real numbers, and n must be a whole number.

Examples that ARE polynomials:

• x² + 3x + 7
• 3x⁴ + 7x² + 9

Examples that are NOT polynomials:

• 2x² + 1/x² (because 1/x² becomes x⁻², and −2 is not a whole number)
• 3x² + √x (because √x is x^(1/2), and 1/2 is not a whole number)

An algebraic expression with one variable is called univariate, with two variables bivariate, and with three variables trivariate.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Degree can never be negative, since powers must be whole numbers.

Example 1: In 8x⁷ + 3x² + 9x⁵ + 3x⁷, the highest power is 8 (from 8x⁷), so the degree is 8.

Example 2: In 5y³ + y² + 2x²y − 1, the highest power is 3, so the degree is 3.

A Common Mistake — Cancelling Terms

In the expression 12x⁷ + 3x⁴ + 2x⁵ + 9x² + 4x³ − 12x⁷, many students answer "degree 7" because 7 looks like the biggest power. This is wrong. Since +12x⁷ and −12x⁷ cancel each other out, the actual highest surviving power is 5, so the correct degree is 5.

Coefficients and Constants

In 6x² − 3x³ + …, the coefficient of x² is 6 and the coefficient of x³ is −3 (the negative sign must be included). In an expression ending in "−10", the constant term is −10, not 10 — always carry the sign.

Types of Polynomials by Degree

Quick check: The degree of 9x³ + 5x² − 8x − 10 is 3, since x³ has the highest power. Among x², 4z − 3, x³, and a constant term, only 4z − 3 is a linear polynomial, since its highest power is 1.

What Is a Linear Polynomial?

A linear polynomial is an algebraic expression whose highest power (degree) is exactly 1. The graph of any linear polynomial is always a straight line.

Example: The perimeter of a square with side x is x + x + x + x = 4x — a linear polynomial.

Example 5: Chess Club Joining Fee

A chess club charges a joining fee of ₹200, plus ₹50 for every match played. If the number of matches is taken as x, the total fee is:

Fee = 200 + 50x

Linear Polynomial as an Input–Output Machine

A linear polynomial can be thought of as a machine: put in a value of x, and a value of y comes out. For y = 2x + 3:

This is written in standard form as y = px, where p(x) represents the polynomial.

Practice: Substituting a Negative Value

For 5x − 3, substituting x = −1 gives 5(−1) − 3 = −5 − 3 = −8.

Assertion–Reason Check

Assertion: 2x + 3 is a linear polynomial. Reason: The highest power of the variable x is 1. Both statements are true, and the reason correctly explains the assertion.

Linear Polynomials in Word Problems

Salil's Age Problem

Salil's mother's age is three times Salil's age. Let Salil's age = x years, so his mother's age = 3x years.

After 5 years:

• Salil's age = x + 5
• Mother's age = 3x + 5

Given that their ages add up to 70 after 5 years:

(x + 5) + (3x + 5) = 70 4x + 10 = 70 4x = 60 x = 15

So Salil is 15 years old, and his mother is 15 × 3 = 45 years old.

The Fence-Cutting Problem

A farmer cuts a 300-foot fence into two pieces, where the longer piece is four times the shorter piece.

Let the shorter piece = x feet, so the longer piece = 4x feet.

x + 4x = 300 5x = 300 x = 60

The shorter piece is 60 feet, and the longer piece is 4 × 60 = 240 feet.

The Ratio Problem

The difference between two positive integers is 63, and their ratio is 2:5.

Let the integers be 2x and 5x.

5x − 2x = 63 3x = 63 x = 21

The two integers are 2 × 21 = 42 and 5 × 21 = 105.

Exploring Linear Patterns

Some sequences increase by a constant amount at each stage — for example, 1, 3, 5, 7, 9, where the difference between consecutive terms is always 2. This pattern can be written as the linear expression 2x − 1, where substituting x = 1, 2, 3, 4 gives 1, 3, 5, 7.

Pocket Money Example

A student has ₹100 pocket money and spends ₹5 every day. After n days, the amount spent is 5n, so the remaining amount is:

Remaining money = 100 − 5n

To find when ₹40 is left:

40 = 100 − 5n −60 = −5n n = 12

So ₹40 remains after 12 days.

Auto Rickshaw Fare Example

An auto rickshaw charges a fixed ₹25 for the first 2 km, and then ₹15 for every kilometre after that. For a 10 km trip, the first 2 km cost ₹25, and the remaining 8 km are charged at ₹15 per km. Working through the calculation as shown in the chapter, the total fare for a 10 km ride comes to ₹115.

Linear Growth and Linear Decay

When a quantity increases by a constant amount over equal intervals, it is represented by a straight line with a positive slope — this is called linear growth. When a quantity decreases by a constant amount over equal intervals, it is represented by a straight line with a negative slope — this is called linear decay.

• A cycle's resale value dropping every month is linear decay (negative slope).
• Pocket money increasing by a fixed amount every week is linear growth (positive slope).

Journey Cost Example (Linear Growth)

A journey costs a fixed ₹100, plus ₹60 for every kilometre travelled. The cost for each kilometre keeps increasing:

Since the cost keeps rising, the graph slopes upward — this is linear growth.

Water Tank Example (Linear Decay)

A tank's water height follows h(t) = 3 − 0.5t, where t is time in months.

As t increases, height decreases — this is linear decay, shown by a negative slope.

Mobile Phone Depreciation Example

A mobile phone bought for ₹10,000 loses ₹800 in value every year. Since the value keeps falling, this is also an example of linear decay, not linear growth.

Assertion–Reason: Population Model

Assertion: The population model P(T) = 750 + 50T shows linear growth (since population increases from 800 at T = 1 to 850 at T = 2). Reason: A straight line with a negative slope represents linear growth.

The assertion is true, but the reason is false — a negative slope represents linear decay, not growth, and this particular line has a positive slope.

Plant Growth Example

A plant is 1.75 feet tall and grows by 0.5 feet every month. The height after t months is:

Height = 1.75 + 0.5t

After 7 months:

1.75 + 0.5(7) = 1.75 + 3.5 = 5.25 feet

The Linear Relationship y = ax + b

A linear relationship between x and y is written as y = ax + b (also seen as y = mx + c), where:

• a (or m) is the slope
• b (or c) is the y-intercept — the point where the line crosses the y-axis

Example: In y = 2x − 1, the slope is 2 and the y-intercept is −1. Substituting x = 0 gives y = −1, meaning the line crosses the y-axis below zero.

Telephone Bill Example

A telephone company charges a fixed monthly fee plus a cost per GB of data, modelled as y = ax + b.

• When x = 10 GB, the bill y = ₹350 → 350 = 10a + b
• When x = 20 GB, the bill y = ₹550 → 550 = 20a + b

Subtracting the first equation from the second:

(20a + b) − (10a + b) = 550 − 350 10a = 200 a = 20

Substituting back: 10(20) + b = 350 → 200 + b = 350 → b = 150

Practice: Gym / Badminton Court Charges

Using the same y = ax + b model, a badminton court charges a fixed fee plus an hourly rate:

• 10 hours of use → bill of ₹800
• 15 hours of use → bill of ₹1,100

Set up two equations, 10a + b = 800 and 15a + b = 1,100, subtract them to find a (cost per hour) and b (fixed charge), then use those values to predict the bill for 20 hours of use. Try solving this yourself using the same subtraction method shown in the telephone example above.

Visualising Linear Relationships on a Graph

Any linear equation y = ax + b can be drawn as a straight line, and only two points are needed to plot it.

Example: For y = 2x + 1:

• At x = 0, y = 1 → point (0, 1)
• At x = 3, y = 7 → point (3, 7)

Joining these two points gives the straight-line graph of y = 2x + 1.

How the Slope (a) Affects the Graph

Comparing y = x, y = 2x, and y = x/2, all pass through the origin (0, 0) since b = 0, but the steepness changes with a:

• A larger value of a makes the line steeper.
• A smaller value of a makes the line flatter.

How the Y-Intercept (b) Affects the Graph

Comparing y = x + 3, y = x, and y = x − 3, all three lines are parallel to each other because they share the same slope (1) — only the point where they cross the y-axis changes. Lines are parallel to each other when they have the same slope.

Practice Questions

• For y = −3x + 4, the slope is −3 and the y-intercept is 4.
• Given 2y = 4x + 7 and 3y = 6x − 11, dividing both equations gives y = 2x + 7/2 and y = 2x − 11/2. Since both have slope 2, the two lines are parallel.

Quick Revision Round

• The degree of a constant polynomial is always 0.
• A number multiplied by 5 and then added to 21 equals 96 is written as the linear equation 5x + 21 = 96.
• For y = −5x, the slope is −5 — true. The coefficient of x in y = ax is called the slope — also true, and it correctly explains the first statement.