INTRODUCTION TO
LINEAR POLYNOMIALS
From Algebraic Expressions to Straight Lines on the Coordinate Plane
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Our Learning Journey
1. Polynomials
Univariate • Degree • Types
2. Linear Polynomial
Definition • General Form
3. Linear Patterns
Constant Difference
4. Growth & Decay
With Graphs
5. y = ax + b
Slope & Intercept
6. Visualising
Straight Lines
All content is taken directly from Ganita Manjari Class 9, Chapter 2
What is a Univariate Polynomial?
An algebraic expression that involves only one variable and the non-negative integer powers of that variable is called a univariate polynomial (or simply a polynomial).
Examples
3x + 7 • x² + 5x + 1
5y³ + y² + 2y – 1
Degree
The highest power of the variable in a polynomial is called its degree.
Classification by Degree
Degree 0
Constant
Polynomial
8 = 8x⁰
Degree 1
Linear
Polynomial
3z + 7
Degree 2
Quadratic
Polynomial
x² + 5x + 1
Degree 3
Cubic
Polynomial
5y³ + y² – 1
In this chapter we focus only on Linear Polynomials (degree 1).
Linear Polynomial
A polynomial of degree one is called a Linear Polynomial.
General Form
ax + b
(where a ≠ 0)
Here a is the coefficient of the variable and b is the constant term.
What is a Linear Pattern?
A sequence of numbers in which the difference between consecutive terms is constant is called a linear pattern.
Example from the book (Square tiles):
1 , 3 , 5 , 7 , 9 , …
Constant difference = 2
Polynomial as an Input-Output Process
This process is later called a function.
Linear Growth
Example: Cost of journey C(d) = 100 + 60d
Every extra kilometre adds a constant ₹60.
Graph of linear growth is a straight line with positive slope.
Linear Decay
Example: Height of water h(t) = 3 – 0.5t
Every month the height falls by a constant 0.5 m.
Graph of linear decay is a straight line with negative slope.
The Equation of a Straight Line
Any linear relationship between two variables x and y can be written as
y = ax + b
a → Slope
Rate of change
b → Y-Intercept
Value when x = 0
What does Slope (a) tell us?
a > 0
Line goes upward
from left to right
Linear Growth
a < 0
Line goes downward
from left to right
Linear Decay
|a| large
Line becomes
steeper
Greater change
Slope also represents the constant difference in a linear sequence.
What is the Y-Intercept (b)?
y = 2x + 5
b = 5 → cuts at (0, 5)
y = 3x – 2
b = –2 → cuts at (0, –2)
When are two lines Parallel?
Example:
y = 2x – 1
y = 2x + 1
y = 2x + 5
All have slope a = 2 → parallel to each other.
Standard Procedure to Plot y = ax + b
“A straight line is uniquely determined by two points.”
Key Ideas to Remember
• Degree = highest power
• Linear decay → negative slope
• b = y-intercept (0, b)
Thank You
“Mathematics is the language in which God has written the universe.”
— Galileo Galilei
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)