Class 9 • Mathematics • Ganita Manjari
📐 Introduction to Linear Polynomials
Chapter 2 — A complete, student-friendly guide with definitions, examples, patterns, and exam practice
Polynomials
Linear Equations
Linear Patterns
Growth & Decay
Graphs
Slope & Intercept
Linear Equations
Linear Patterns
Growth & Decay
Graphs
Slope & Intercept
📋 Contents
2.1 Introduction to Algebraic Expressions
An algebraic expression is a combination of numbers, variables (letters representing unknown values), and operation symbols (+, –, ×, ÷).
Real-Life Story: Raju at the Shop
Raju buys x red boxes (4 pens each) and y blue boxes (5 pencils each), and also gets 3 free pens. The total is expressed as the algebraic expression 4x + 5y + 3.
Raju buys x red boxes (4 pens each) and y blue boxes (5 pencils each), and also gets 3 free pens. The total is expressed as the algebraic expression 4x + 5y + 3.
🔑 Key Terms in an Algebraic Expression
Terms
Parts separated by + or – signs. In 4x + 5y + 3, the terms are 4x, 5y, and 3.
Parts separated by + or – signs. In 4x + 5y + 3, the terms are 4x, 5y, and 3.
Variables
The letter symbols whose values can change. Here, x and y are variables.
The letter symbols whose values can change. Here, x and y are variables.
Coefficients
Numbers multiplied with variables. 4 is the coefficient of x; 5 is the coefficient of y.
Numbers multiplied with variables. 4 is the coefficient of x; 5 is the coefficient of y.
Constant
A fixed number with no variable. In 4x + 5y + 3, the constant is 3.
A fixed number with no variable. In 4x + 5y + 3, the constant is 3.
Remember!
“Letter-numbers” are officially called variables. From now on, always use the word variable in your exams!
“Letter-numbers” are officially called variables. From now on, always use the word variable in your exams!
📐 Another Real-Life Example: Rectangular Garden
A garden of length l m and width w m has different fencing and seeding costs. The total cost works out to the expression 200l + 160w + 50lw.
w metres (wooden fence ₹80/m)
w metres
l metres (wire ₹100/m)
l metres
Seeds ₹50/m²
(area = l × w)
Total Cost = 200l + 160w + 50lw
Note
Expressions like 4x + 5y + 3 have two variables. In this chapter, we focus only on expressions with one variable.
Expressions like 4x + 5y + 3 have two variables. In this chapter, we focus only on expressions with one variable.
Polynomials — Definition & Types
Algebraic expressions involving one variable and its powers are called univariate polynomials (or simply polynomials).
Degree of a Polynomial = Highest power of the variable
📊 Types of Polynomials by Degree
| Type | Degree | Example |
|---|---|---|
| Constant Polynomial | 0 | 8 = 8x⁰ |
| Linear Polynomial | 1 | 3z + 7 |
| Quadratic Polynomial | 2 | x² + 5x + 1 |
| Cubic Polynomial | 3 | 5y³ + y² + 2y − 1 |
Quick Tip to Find Degree
Ignore the coefficients — just find the highest exponent of the variable. In 5y³ + y² + 2y − 1, the highest power is 3, so it’s a cubic polynomial.
Ignore the coefficients — just find the highest exponent of the variable. In 5y³ + y² + 2y − 1, the highest power is 3, so it’s a cubic polynomial.
🔍 Example: Identify Parts of a Polynomial
Polynomial: 5y³ + y² + 2y − 1Coefficient of y³ → 5
Coefficient of y² → 1
Coefficient of y → 2
Constant term → −1
Degree → 3 (cubic)
Coefficient of y² → 1
Coefficient of y → 2
Constant term → −1
Degree → 3 (cubic)
Common Mistake!
x⁴ − 3x³ + 6x² − 2x + 7 — the coefficient of x² is 6 and of x³ is −3 (not +3). Always include the sign!
x⁴ − 3x³ + 6x² − 2x + 7 — the coefficient of x² is 6 and of x³ is −3 (not +3). Always include the sign!
2.2 Linear Polynomials & Linear Equations
A polynomial of degree 1 is called a linear polynomial. It has the general form ax + b, where a ≠ 0.
Linear Polynomial: ax + b (where a ≠ 0)
📌 Real-Life Examples of Linear Polynomials
📐 Square Perimeter
Perimeter of a square with side x = 4x — a linear polynomial in x.
Perimeter of a square with side x = 4x — a linear polynomial in x.
♟ Chess Club Fee
Joining fee ₹200 + ₹50 per match = 200 + 50m — a linear polynomial in m.
Joining fee ₹200 + ₹50 per match = 200 + 50m — a linear polynomial in m.
Key Feature of Linear Polynomials
When you calculate values of a linear polynomial at consecutive integers, the difference is always constant. This is the defining feature of a linear pattern!
When you calculate values of a linear polynomial at consecutive integers, the difference is always constant. This is the defining feature of a linear pattern!
⚖️ Linear Equations
When a linear polynomial is set equal to a constant, we get a linear equation.
Example 6: Finding Two Numbers
Two numbers add up to 64, and one is 10 more than the other. Let the smaller number be x. Then:x + (x + 10) = 64 → 2x + 10 = 64 → 2x = 54 → x = 27
The two numbers are 27 and 37.
Two numbers add up to 64, and one is 10 more than the other. Let the smaller number be x. Then:x + (x + 10) = 64 → 2x + 10 = 64 → 2x = 54 → x = 27
The two numbers are 27 and 37.
⚙️ Polynomials as Input–Output Machines
Think of a polynomial as a function: you put in a value of x (input), and the polynomial gives you an output.
x = 4
y = 2x + 3
( Linear Function )
y = 11
(2×4+3=11)
Linear vs Quadratic Functions
2x + 3 is a linear function. 10x − x² is a quadratic function (degree 2).
2x + 3 is a linear function. 10x − x² is a quadratic function (degree 2).
2.3 Exploring Linear Patterns
A linear pattern is a sequence of numbers where the difference between any two consecutive terms is constant.
Linear Pattern: nth term = an + b
🔷 Growing Pattern of Square Tiles
| Stage (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Number of Tiles | 1 | 3 | 5 | 7 | 9 | 11 | 13 |
Number of tiles at Stage n = 2n − 1
The difference between consecutive terms is always 2 — a constant. So this is a linear pattern!
Verification
Stage 5: 2(5) − 1 = 9 ✓ | Stage 15: 2(15) − 1 = 29 tiles | Which stage has 47 tiles? → 2n−1 = 47 → n = 24
Stage 5: 2(5) − 1 = 9 ✓ | Stage 15: 2(15) − 1 = 29 tiles | Which stage has 47 tiles? → 2n−1 = 47 → n = 24
💰 Example 7: Bela’s Pocket Money
Problem
Bela has ₹100. She spends ₹5 each day. After how many days does she have ₹40?Amount on day n = 100 − 5n
Set equal to 40: 100 − 5n = 40 → 5n = 60 → n = 12
After 12 days, she has ₹40.
Bela has ₹100. She spends ₹5 each day. After how many days does she have ₹40?Amount on day n = 100 − 5n
Set equal to 40: 100 − 5n = 40 → 5n = 60 → n = 12
After 12 days, she has ₹40.
🚖 Example 8: Auto-Rickshaw Fare
Starting fare ₹25 for first 2 km, then ₹15 per km. For n km (where n ≥ 2):
Fare = 25 + 15(n − 2) = 15n − 5
Fare for 10 km = 15(10) − 5 = ₹145
2.4 Linear Growth and Linear Decay
📈 Linear Growth
A quantity increases by a constant amount over equal intervals. Represented by a line with positive slope.
A quantity increases by a constant amount over equal intervals. Represented by a line with positive slope.
📉 Linear Decay
A quantity decreases by a constant amount over equal intervals. Represented by a line with negative slope.
A quantity decreases by a constant amount over equal intervals. Represented by a line with negative slope.
🚗 Example 9: Journey Cost (Linear Growth)
C(d) = 100 + 60d
Where C = total cost (₹), d = distance (km). Each extra km adds ₹60.
| Distance d (km) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cost C (₹) | 100 | 160 | 220 | 280 | 340 | 400 |
Cost increases by ₹60 per km — constant increase → Linear Growth
💧 Example 10: Water Tank (Linear Decay)
h(t) = 3 − 0.5t
Where h = height of water (m), t = months. Water level drops 0.5 m per month.
| Month (t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Height h (m) | 3 | 2.5 | 2 | 1.5 | 1 | 0.5 |
Height decreases by 0.5 m per month — constant decrease → Linear Decay
🏭
Did You Know?
Linear growth and decay models are used in real life for depreciation of assets (phones, cars), population growth, radioactive decay estimates, and even monthly EMI planning!
Linear growth and decay models are used in real life for depreciation of assets (phones, cars), population growth, radioactive decay estimates, and even monthly EMI planning!
2.5 Linear Relationships
A linear relationship between two variables x and y is expressed as:
y = ax + b
Here, a is the slope (rate of change) and b is the y-intercept (constant).
📡 Example 11: Telecom Bill (Finding a and b)
Given Information
When data used = 10 GB → Bill = ₹350 | When data used = 20 GB → Bill = ₹550
Find a and b in y = ax + b
When data used = 10 GB → Bill = ₹350 | When data used = 20 GB → Bill = ₹550
Find a and b in y = ax + b
Step 1: Write two equations from given data:
350 = 10a + b … (i)
550 = 20a + b … (ii)Step 2: From (i): b = 350 − 10a
Step 3: Substitute in (ii): 550 = 20a + (350 − 10a)
550 = 10a + 350
10a = 200 → a = 20Step 4: b = 350 − 10(20) = 350 − 200 = b = 150
350 = 10a + b … (i)
550 = 20a + b … (ii)Step 2: From (i): b = 350 − 10a
Step 3: Substitute in (ii): 550 = 20a + (350 − 10a)
550 = 10a + 350
10a = 200 → a = 20Step 4: b = 350 − 10(20) = 350 − 200 = b = 150
Answer: y = 20x + 150
(₹20 per GB, ₹150 fixed monthly fee)
Interpretation
In y = 20x + 150: 20 is the cost per GB (rate), and 150 is the fixed monthly charge. In any y = ax + b, a = rate of change and b = starting value.
In y = 20x + 150: 20 is the cost per GB (rate), and 150 is the fixed monthly charge. In any y = ax + b, a = rate of change and b = starting value.
2.6 Visualising Linear Relationships (Graphs)
Every linear equation y = ax + b represents a straight line on the coordinate plane.
📌 Steps to Plot a Linear Graph
- Choose any two values of x (tip: use x = 0 and another convenient value).
- Calculate the corresponding y values using the equation.
- Plot the two points on the coordinate plane.
- Join the points with a straight line and extend it in both directions.
📐 Example: Plot y = 2x + 1
Point A (x = 0)
y = 2(0) + 1 = 1 → A(0, 1)
y = 2(0) + 1 = 1 → A(0, 1)
Point B (x = 3)
y = 2(3) + 1 = 7 → B(3, 7)
y = 2(3) + 1 = 7 → B(3, 7)
x
y
1
2
3
-1
1
3
5
-2
A(0,1)
B(3,7)
y = 2x+1
🔑 Key Properties of y = ax + b
| Parameter | Role | Example |
|---|---|---|
| a (slope) | Steepness and direction of the line | In y = 3x + 2, slope = 3 |
| b (y-intercept) | Where the line crosses the y-axis | In y = 3x + 2, line cuts y-axis at (0,2) |
| a > 0 | Line goes up (left to right) → Linear Growth | y = 2x + 1 |
| a < 0 | Line goes down (left to right) → Linear Decay | y = −3x + 5 |
| Equal a, different b | Lines are parallel to each other | y=2x−1, y=2x+3 are parallel |
| b = 0 | Line passes through the origin (0,0) | y = 3x |
Three Important Conclusions (from Chapter)
(i) In y = ax + b: a = slope, b = y-intercept.
(ii) Changing a (keeping b fixed) → changes slope, but y-intercept stays.
(iii) Changing b (keeping a fixed) → parallel lines with same slope.
(i) In y = ax + b: a = slope, b = y-intercept.
(ii) Changing a (keeping b fixed) → changes slope, but y-intercept stays.
(iii) Changing b (keeping a fixed) → parallel lines with same slope.
Exam Tip: Steepness
Lines of the form y = ax always pass through origin. When a > 1, the line is steeper than y = x. When a < 1, it is less steep.
Lines of the form y = ax always pass through origin. When a > 1, the line is steeper than y = x. When a < 1, it is less steep.
Chapter Summary
📝 Algebraic Expression
Combines numbers, variables & operations. Terms, variables, coefficients, and constants are its parts.
Combines numbers, variables & operations. Terms, variables, coefficients, and constants are its parts.
📐 Polynomial Degree
The highest power of the variable. Degree 0 = constant, 1 = linear, 2 = quadratic, 3 = cubic.
The highest power of the variable. Degree 0 = constant, 1 = linear, 2 = quadratic, 3 = cubic.
📏 Linear Polynomial
Degree 1 expression: ax + b (a ≠ 0). Examples: 2x + 3, 5 − 4y.
Degree 1 expression: ax + b (a ≠ 0). Examples: 2x + 3, 5 − 4y.
📈 Linear Growth
Quantity increases by a fixed amount each interval. Positive slope on graph.
Quantity increases by a fixed amount each interval. Positive slope on graph.
📉 Linear Decay
Quantity decreases by a fixed amount each interval. Negative slope on graph.
Quantity decreases by a fixed amount each interval. Negative slope on graph.
🔗 Linear Relationship
y = ax + b. Slope = a. y-intercept = b. Parallel lines share same slope.
y = ax + b. Slope = a. y-intercept = b. Parallel lines share same slope.
Quick Formula Sheet
• General form: y = ax + b | • Linear pattern nth term: an + b
• y-intercept = b (set x = 0) | • Parallel lines: same slope a, different b
• Lines through origin: y = ax (b = 0)
• General form: y = ax + b | • Linear pattern nth term: an + b
• y-intercept = b (set x = 0) | • Parallel lines: same slope a, different b
• Lines through origin: y = ax (b = 0)
Exam Practice Questions
📝 Short Answer (1–2 Marks)
Q1. Find the degree of the polynomial 2x² − 5x + 3.
Ans: The highest power of x is 2. So, the degree is 2 (Quadratic polynomial).
Q2. What is the coefficient of x² and x³ in x⁴ − 3x³ + 6x² − 2x + 7?
Ans: Coefficient of x² = 6; Coefficient of x³ = −3.
Q3. Find the value of the linear polynomial 5x − 3 when x = 2.
Ans: 5(2) − 3 = 10 − 3 = 7.
Q4. Which stage of the tile pattern (2n − 1) will have 47 tiles?
Ans: 2n − 1 = 47 → 2n = 48 → n = 24th stage.
🧮 Application Questions (3–4 Marks)
Q5. The sum of two numbers is 64. One number is 10 more than the other. Find both numbers.
Ans: Let smaller = x. Then x + (x+10) = 64 → 2x = 54 → x = 27. Numbers are 27 and 37.
Q6. A plant is 1.75 feet tall and grows 0.5 feet per month. Write the linear expression for height after t months. Find the height after 7 months.
Ans: h(t) = 1.75 + 0.5t. After 7 months: h(7) = 1.75 + 3.5 = 5.25 feet. This is linear growth.
Q7. A mobile phone worth ₹10,000 depreciates by ₹800/year. Find its value after 3 years and write the linear decay expression.
Ans: v(t) = 10000 − 800t. After 3 years: v(3) = 10000 − 2400 = ₹7,600. (Linear decay: negative slope)
Q8. A learning app costs ₹400 for 10 modules and ₹500 for 14 modules. Find the relation y = ax + b.
Ans: 400 = 10a + b and 500 = 14a + b. Subtracting: 100 = 4a → a = 25. b = 400 − 250 = 150. So, y = 25x + 150.
📊 Graph-Based Questions (5 Marks)
Q9. Draw the graph of y = 2x + 1. Find its slope and y-intercept. Where does it cut the y-axis?
Ans: Slope = 2, y-intercept = 1. The line cuts the y-axis at (0, 1). Plot points (0,1) and (3,7) and join them.
Q10. The graph of a linear polynomial p(x) passes through (1, 5) and (3, 11). Find p(x).
Ans: 5 = a + b and 11 = 3a + b. Subtracting: 6 = 2a → a = 3. b = 5 − 3 = 2. So, p(x) = 3x + 2.
Common Exam Mistakes to Avoid
① Never forget the negative sign in coefficients (e.g., −3x has coefficient −3, not 3).
② When finding degree, look at the variable’s power, NOT the coefficient.
③ For parallel lines, slopes must be EQUAL but y-intercepts must be DIFFERENT.
④ Always verify a point lies on a line by substituting both coordinates in the equation.
① Never forget the negative sign in coefficients (e.g., −3x has coefficient −3, not 3).
② When finding degree, look at the variable’s power, NOT the coefficient.
③ For parallel lines, slopes must be EQUAL but y-intercepts must be DIFFERENT.
④ Always verify a point lies on a line by substituting both coordinates in the equation.
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