Class 9 Mathematics · Chapter 2 NotesIntroduction to Linear Polynomials
Learn about linear polynomials, their values, operations, applications, and visualizations. Understand algebraic expressions, variables, and polynomials with these…
In this chapter, we explore a special type of algebraic expression called linear polynomials. These are expressions with one variable and a highest power of 1. You'll learn how to understand, evaluate, and work with linear polynomials, and see how they model real-world situations. By the end, you'll be able to recognize linear patterns, understand linear growth and decay, and visualize linear relationships through graphs.
What you'll learn
1Understand what algebraic expressions and variables are
2Identify and define polynomials and their degrees
3Recognize and work with linear polynomials
4Perform operations with linear polynomials
5Apply linear polynomials to real-world situations
6Understand and interpret linear growth and decay
7Visualize linear relationships through graphs
Chapter at a glance
01Algebraic Expressions and Variables
02Understanding Polynomials
03Operations on Linear Polynomials
04Applications of Linear Polynomials
Detailed chapter notes
01
Algebraic Expressions and Variables
Algebraic expressions combine numbers, variables (like x, y, or z), and operation symbols. For example, 4x + 5y + 3 is an algebraic expression where 4x, 5y, and 3 are terms, x and y are variables, and 4 and 5 are coefficients. The number 3 is a constant. Variables can represent changing values, and coefficients are the numbers that multiply the variables.
Algebraic expressions combine numbers, variables, and operation symbols
Variables represent changing values (e.g., x, y, z)
Coefficients are the numbers that multiply the variables (e.g., 4 in 4x)
Constants are terms without variables (e.g., 3 in 4x + 3)
02
Understanding Polynomials
Polynomials are algebraic expressions with one variable and its powers. The highest power of the variable in a polynomial is called its degree. For example, 5y³ + y² + 2y – 1 is a polynomial of degree 3 (cubic polynomial), while 3z + 7 is a polynomial of degree 1 (linear polynomial). The constant 8 is a polynomial of degree 0 (constant polynomial).
Polynomials are algebraic expressions with one variable and its powers
Degree of a polynomial is the highest power of the variable
Linear polynomials are polynomials of degree 1, meaning they have only the first power of the variable. Examples include 2x + 3 and 5 – 4y. To find the value of a linear polynomial for a given value of the variable, substitute the value into the polynomial. For instance, if x = 4 in 2x + 3, the value is 2(4) + 3 = 11. Linear polynomials can model real-world situations, like calculating the perimeter of a square or the total cost of an item with a fixed fee plus a variable charge.
Linear polynomials have degree 1 (e.g., 2x + 3, 5 – 4y)
To find the value of a linear polynomial, substitute the variable's value
ExampleFor x = 4 in 2x + 3, the value is 2(4) + 3 = 11
Linear polynomials can model real-world situations
04
Operations on Linear Polynomials
You can perform operations like addition, subtraction, and multiplication with linear polynomials. To add or subtract linear polynomials, combine like terms. For example, (3x + 2) + (x – 4) = 4x – 2. To multiply linear polynomials, use the distributive property. For instance, (2x + 3)(x – 1) = 2x² + x – 3. These operations help simplify expressions and solve equations involving linear polynomials.
Addition and subtractionCombine like terms (e.g., (3x + 2) + (x – 4) = 4x – 2)
MultiplicationUse the distributive property (e.g., (2x + 3)(x – 1) = 2x² + x – 3)
Operations help simplify expressions and solve equations
05
Applications of Linear Polynomials
Linear polynomials have many real-world applications. They can model situations involving linear growth or decay, where a quantity increases or decreases by a constant amount over equal intervals. For example, a plant growing by 0.5 feet each month or a phone's value decreasing by `800 every year. Linear polynomials can also represent linear relationships between two variables, like the relationship between the number of matches played and the total cost in a chess club.
Model linear growth (e.g., plant growing by 0.5 feet each month)
Model linear decay (e.g., phone's value decreasing by `800 every year)
Represent linear relationships between two variables (e.g., cost vs. matches played)
Help solve real-world problems involving constant rates of change
06
Visualizing Linear Relationships
Linear relationships can be visualized through graphs. A linear relationship between two variables x and y is represented by a straight line y = ax + b. The slope of the line is a, and the y-intercept is b, the point where the line crosses the y-axis. For example, the line y = 2x + 1 has a slope of 2 and a y-intercept at (0, 1). Graphs help understand and interpret linear relationships, like how the number of square tiles in a pattern increases with each stage.
Linear relationships are represented by straight lines (e.g., y = ax + b)
Slope (a) represents the rate of change
Y-intercept (b) is the point where the line crosses the y-axis
Graphs help understand and interpret linear relationships
Want the complete chapter resources?Topic notes, quizzes and flashcards for Introduction to Linear Polynomials.
Q1. Define a linear polynomial. Give two examples of linear polynomials in one variable and identify their terms, coefficients, and constant terms.
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Model answer
A linear polynomial is a polynomial of degree 1. Examples: 3x + 7 and 2y - 5. In 3x + 7, the terms are 3x and 7; the coefficient of x is 3, and the constant term is 7. In 2y - 5, the terms are 2y and -5; the coefficient of y is 2, and the constant term is -5.
Sample question3 marks
Q2. Define a polynomial. Give two examples of polynomials in one variable, and identify the degree of each.
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Model answer
A polynomial is an algebraic expression involving one variable and its non-negative integer powers. For example, 3z + 7 is a polynomial in z of degree 1, and x^2 + 5x + 1 is a polynomial in x of degree 2. The degree is the highest power of the variable in the polynomial.
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Q3. Define a linear polynomial. Give two examples and find the degree of each.
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Model answer
A linear polynomial is a polynomial of degree one. It is of the form ax + b, where a and b are constants and a ≠ 0. Examples: 3z + 7 and 5 – 4y. The degree of 3z + 7 is 1, and the degree of 5 – 4y is also 1.
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Q4. Find the value of the linear polynomial 5x - 3 when x = 0, x = -1, and x = 2.
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Model answer
To evaluate the linear polynomial 5x - 3, substitute the given values of x. For x = 0, 5(0) - 3 = -3. For x = -1, 5(-1) - 3 = -5 - 3 = -8. For x = 2, 5(2) - 3 = 10 - 3 = 7. Thus, the values are -3, -8, and 7 respectively.
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Q5. A cab service charges a fixed amount of ₹25 for the first 2 km and then ₹15 per km for every additional kilometre. Write a linear polynomial to represent the fare for a journey of n km (n ≥ 2) and find the fare for a 10 km journey.
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Model answer
For n km, the fare is ₹25 for the first 2 km and ₹15 for each additional (n – 2) km. So the fare is 25 + 15(n – 2) = 25 + 15n – 30 = 15n – 5. For n = 10, fare = 15×10 – 5 = 150 – 5 = ₹145.
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A linear polynomial is an algebraic expression with one variable and a highest power of 1. Examples include 2x + 3 and 5 – 4y.
How do you find the value of a linear polynomial?
To find the value of a linear polynomial, substitute the given value of the variable into the polynomial and simplify. For example, if x = 4 in 2x + 3, the value is 2(4) + 3 = 11.
What is the difference between linear growth and linear decay?
Linear growth is a pattern where a quantity increases by a constant amount over equal intervals, while linear decay is a pattern where a quantity decreases by a constant amount over equal intervals.
How do you visualize a linear relationship?
A linear relationship can be visualized through a graph, which is a straight line represented by the equation y = ax + b. The slope (a) represents the rate of change, and the y-intercept (b) is the point where the line crosses the y-axis.
What are some real-world applications of linear polynomials?
Linear polynomials can model situations involving linear growth or decay, like a plant growing by a constant amount each month or a phone's value decreasing by a constant amount each year. They can also represent linear relationships between two variables, like the relationship between the number of matches played and the total cost in a chess club.
What is the degree of a polynomial?
The degree of a polynomial is the highest power of the variable in the polynomial. For example, the degree of 5y³ + y² + 2y – 1 is 3, and the degree of 3z + 7 is 1.