📝 Section 1 — Classroom Notes
🌎 Real-World Hook: Imagine you start a savings jar with ₹200, and every week you add ₹50. After x weeks, your total is 200 + 50x . This is a linear polynomial ! Phone bills, taxi fares, and school canteen tabs all follow this pattern — a fixed charge plus a changing amount. Polynomials are the language of any situation where things grow (or shrink) in a predictable, straight-line way.
🧩 What is an Algebraic Expression?
It mixes numbers and letters with +, -, x.
Example: 4x + 5y + 3 .
The letters (x, y) are called variables .
The numbers (4, 5) are called coefficients .
A plain number (3) is called a constant .
Each piece (4x, 5y, 3) is called a term .
📦 What is a Polynomial?
A polynomial uses one variable and its powers.
Example: 5y³ + y² + 2y - 1 .
The highest power is called the degree .
Only whole number powers are used (0, 1, 2, 3...).
🏷️ Types of Polynomials (by Degree)
Degree 0 = Constant (just a number, like 8).Degree 1 = Linear (like 3x + 7).Degree 2 = Quadratic (like x² + 5x + 1).Degree 3 = Cubic (like 5y³ + y² - 1).The degree = the biggest power of the variable.
⭐ Linear Polynomials
A linear polynomial has degree 1 .
It looks like ax + b .
"a" is the coefficient of x.
"b" is the constant .
Examples: 2x + 3, 5 - 4y, 50m + 200.
The graph is always a straight line .
🔢 Evaluating a Polynomial
Evaluating means putting in a number for x.Think of it like an input-output machine .
You put a number in , the formula gives a number out .
Example: for 2x + 3 , put in x = 4.
Output = 2 x 4 + 3 = 11 .
Put in x = -6. Output = 2 x (-6) + 3 = -9 .
📈 Linear Patterns
A linear pattern is a list of numbers.
The difference between neighbours is always the same .
Example: 1, 3, 5, 7, 9... (difference = 2 each time).
We can write a formula for the pattern.
Here the formula is 2n - 1 (n = term number).
📈 Linear Growth
Linear growth = value goes up by the same amount.Example: save ₹150 every month.
Month 1: ₹650, Month 2: ₹800, Month 3: ₹950...
The amount increases by ₹150 each month.
The line on a graph goes up (positive slope).
📉 Linear Decay
Linear decay = value goes down by the same amount.Example: water drops by 0.5 m every month.
Month 0: 3 m, Month 1: 2.5 m, Month 2: 2 m...
The height decreases by 0.5 m each month.
The line on a graph goes down (negative slope).
📏 Slope and Y-intercept
A straight line has the equation y = ax + b .
a is the slope (how steep the line is).b is the y-intercept (where line crosses y-axis).If a is positive , the line goes up (growth).
If a is negative , the line goes down (decay).
Bigger "a" = steeper line.
🔀 Parallel Lines
Parallel lines have the same slope (same "a").But different y-intercepts (different "b").
They never cross each other.
Example: y = 2x + 1 and y = 2x + 5 are parallel.
🎨 Section 2 — Easy Illustrations
Picture 1: Types of Polynomials
Constant
Degree 0
8
just a number
Linear
Degree 1
3x + 7
straight line
Quadratic
Degree 2
x² + 5x + 1
curve (parabola)
Cubic
Degree 3
5y³ + y² - 1
wavy curve
Degree increases →
We study LINEAR
Linear = Degree 1 = highest power of variable is 1
Four types of polynomials by degree. This chapter focuses on linear polynomials (degree 1).
Picture 2: Input-Output Machine
INPUT
x = 4
MACHINE
2x + 3
2 x 4 + 3
= 8 + 3
OUTPUT
11
Evaluating 2x + 3 when x = 4
Put a number in (x = 4), the formula 2x + 3 gives you a number out (11).
Picture 3: Linear Pattern (Tiles)
Stage 1
1 tile
Stage 2
3 tiles
Stage 3
5 tiles
Pattern Rule:
Tiles = 2n - 1
n = stage number
+2
+2
Difference is always 2
= Linear Pattern!
Each stage adds 2 tiles. The constant difference makes it a linear pattern. Formula: 2n - 1.
Picture 4: Growth vs Decay
Linear Growth (goes UP)
time →
value →
↗
slope is positive (+)
Linear Decay (goes DOWN)
time →
value →
↘
slope is negative (-)
Growth: line goes up (positive slope). Decay: line goes down (negative slope).
Picture 5: Slope and Y-intercept
y = ax + b: "a" is slope, "b" is y-intercept
x
y
1
1
2
3
(0, 1) ← y-intercept = b = 1
(3, 7)
run = 3
rise = 6
y = 2x + 1
slope a = 2 (rise / run)
y-intercept b = 1
In y = 2x + 1, the slope is 2 (line rises 2 for every 1 right) and it crosses the y-axis at 1.
✏️ Section 3 — Worked Examples
Example 1: Find the Degree of a Polynomial
Question: What is the degree of 2x² - 5x + 3?
1 Look at all the powers of x: x² has power 2 , x has power 1 , 3 has power 0 .
2 Find the highest power: 2 .
3 The degree is 2 . This is a quadratic polynomial.
Answer: Degree = 2 (Quadratic polynomial)
Example 2: Evaluate a Linear Polynomial
Question: Find the value of 5x - 3 when x = 2.
1 Write the polynomial: 5x - 3 .
2 Replace x with 2: 5 x 2 - 3.
3 Multiply: 5 x 2 = 10 .
4 Subtract: 10 - 3 = 7 .
Answer: When x = 2, the value of 5x - 3 is 7.
Example 3: Find a Linear Relationship y = ax + b
Question: When x = 10, y = 350. When x = 20, y = 550. Find a and b.
1 Write two equations:
2 Subtract (i) from (ii): 550 - 350 = 20a - 10a.a = 20 .
3 Put a = 20 in equation (i): 350 = 10 x 20 + b.b = 150 .
4 Write the final answer: y = 20x + 150 .
Answer: a = 20, b = 150. The relationship is y = 20x + 150.
📊 Section 4 — Cheat Sheet Poster
Linear Polynomials — Cheat Sheet
Chapter 2
🧩 Algebraic Expression
Numbers + letters + operations
e.g. 4x + 5y + 3
terms, variables, coefficients
📦 Polynomial
One variable and its powers
Degree = highest power
degree 0,1,2,3...
🏷️ Polynomial Types
Constant: deg 0
Linear: deg 1
Quadratic: deg 2
Cubic: deg 3
This chapter = LINEAR (degree 1)
⭐ Linear Polynomial
Looks like ax + b
Graph = straight line
e.g. 2x+3, 50m+200
🔢 Linear Patterns
Constant difference between terms
1, 3, 5, 7... (diff = 2 each time)
formula: 2n - 1
📈 Linear Growth
Value goes UP by same amount
Slope is positive (+)
📉 Linear Decay
Value goes DOWN by same amount
Slope is negative (-)
📏 y = ax + b — The Big Formula
a = slope (how steep the line is)b = y-intercept (where line hits y-axis)Same slope + different b = parallel lines
🔀 Parallel Lines
Same slope "a", different "b" = never cross
e.g. y = 2x+1 and y = 2x+5 are parallel (both have slope 2)
Class 9 NCERT - Ganita Manjari - Chapter 2 - Linear Polynomials
📄 Section 5 — Take-Home Handout
Print-friendly. Use Arial or OpenDyslexic font when printing.
🧩 Algebraic Expression
Mixes numbers, letters, and operations (+, -, x).
Example: 4x + 5y + 3 .
Letters = variables . Numbers in front = coefficients .
Example: In 4x + 5y + 3:
1 Terms: 4x , 5y , 3 .
2 Variables: x and y . Coefficients: 4 and 5 . Constant: 3 .
📦 Polynomial and Degree
A polynomial uses one variable and its powers.
Degree = the highest power of the variable.Degree 0 = Constant Linear
Degree 2 = Quadratic Cubic
Example: Degree of 2x² - 5x + 3?
1 Powers: x² = 2, x = 1, 3 = 0.
2 Highest = 2 . So it is a quadratic polynomial.
⭐ Linear Polynomial
Has degree 1 . Looks like ax + b
"a" = coefficient. "b" = constant.
Graph is always a straight line .
Example: Evaluate 5x - 3 when x = 2.
1 Put x = 2: 5 x 2 - 3 = 10 - 3 = 7 .
📈 Linear Growth and 📉 Linear Decay
Growth : value goes UP by same amount each time.Decay : value goes DOWN by same amount each time.Growth = positive slope negative slope
Example (Growth): Save ₹150/month, start with ₹500.
1 Month 1: 500 + 150 = ₹650 .
2 Month n: 500 + 150n (linear growth).
📏 y = ax + b (Slope and Y-intercept)
a = slope (steepness of the line).b = y-intercept (where line crosses y-axis).Same slope, different b = parallel lines
Example: y = 2x + 1
1 Slope a = 2 . Y-intercept b = 1 .
2 Line crosses y-axis at point (0, 1) .
My notes / drawings:
📝 Practice Questions
1. What is the degree of the polynomial 3x³ − 5x² + 7x − 2? __________
2. Evaluate p(x) = 4x − 9 when x = 3. Show your work:
Work space
3. Fill in the blanks:
(a) A polynomial of degree 2 is called a __________ polynomial.
(b) A polynomial of degree 3 is called a __________ polynomial.
(c) The graph of a linear polynomial is always a __________ line.
4. The zero of p(x) = 5x − 15 is x = __________. (Hint: set p(x) = 0 and solve.)
✅ Detailed Solutions — Practice Questions
✓ Question 1: Degree of 3x³ − 5x² + 7x − 2
1 List the powers of x in each term: x³ has power 3 , x² has power 2 , x has power 1 , the constant has power 0 .
2 The highest power is 3.
🏆 Answer: Degree = 3 . This is a cubic polynomial.
✓ Question 2: Evaluate p(x) = 4x − 9 when x = 3
1 Substitute x = 3 into the expression: p(3) = 4 × 3 − 9.
2 Multiply: 4 × 3 = 12 .
3 Subtract: 12 − 9 = 3 .
🏆 Answer: p(3) = 3 .
✓ Question 3: Fill in the Blanks
a Degree 2 = quadratic polynomial.
b Degree 3 = cubic polynomial.
c Graph of linear polynomial = always a straight line.
🏆 Answers: (a) quadratic (b) cubic (c) straight
✓ Question 4: Zero of p(x) = 5x − 15
1 Set p(x) = 0: 5x − 15 = 0.
2 Add 15 to both sides: 5x = 15.
3 Divide both sides by 5: x = 15 ÷ 5 = 3 .
4 Check: p(3) = 5 × 3 − 15 = 15 − 15 = 0. ✓
🏆 Answer: The zero of p(x) = 5x − 15 is x = 3 .
🧠 Section 7 — Mind Map
Linear
Polynomials
Algebraic Expression
numbers + letters + operations
e.g. 4x + 5y + 3
Polynomial
one variable + its powers
Degree Types
0 = Constant, 1 = Linear
2 = Quadratic, 3 = Cubic
ax + b
a = coefficient, b = constant
graph = straight line
Evaluating
put number in, get answer out
Growth and Decay
growth = up (slope +)
decay = down (slope -)
y = ax + b
a = slope (steepness)
b = y-intercept
Parallel Lines
same slope, different b
they never cross
Mind map: all key ideas of Chapter 2 connected together.
Maths Class 9 NCERT Lesson 2 — Linear Polynomials
Special Edition: Different Learner Friendly Learning Materials
Font recommendation: Arial or OpenDyslexic
COLOR VERSION