Chapter 2: Introduction to Linear Polynomials

2.1 Introduction

2.1.1 Algebraic Expressions: Basic Concepts

Algebraic Expressions: Combinations of numbers, letter-numbers (variables), and operation symbols.
Example 1 (Raju's Pens): 4x + 5y + 3.
Terms: Parts of an expression separated by + or - (e.g., 4x, 5y, 3).
Variables: Letter-numbers (e.g., x, y).
Coefficients: Numerical part multiplied by a variable (e.g., 4 in 4x, 5 in 5y).
Constant: Term without a variable (e.g., 3).

2.1.2 Example: Cost in a Rectangular Garden

Problem: Fencing and decorating a garden of length 'l' and width 'w'.
Cost Components: Wire fencing (₹100/m), wooden fencing (₹80/m), seeds (₹50/sq m).
Total Cost Expression: ₹(200l + 160w + 50lw).
Terms, Variables, Coefficients: Can be identified from the expression.

2.1.3 Example: Area of Rectangles from a Wire

Problem: A 20 cm wire bent to form rectangles.
Length & Width: If length is x cm, width is (10 - x) cm (since perimeter is 20 cm, 2(l+w) = 20).
Area Expression: x(10 - x) or 10x - x².
Variable: x.

2.1.4 One-Variable Polynomials and Degree

Restriction: Discussion limited to algebraic expressions with only one variable.
One-Variable Polynomials (Univariate Polynomials): Algebraic expressions involving one variable and its powers (e.g., x² + 5x + 1, 5y³ + y² + 2y - 1).
Degree of a Polynomial: The highest power of the variable in a polynomial.

2.1.4.1 Types of Polynomials by Degree

Cubic Polynomial: Degree 3 (e.g., 5y³ + y² + 2y - 1).
Quadratic Polynomial: Degree 2 (e.g., x² + 5x + 1).
Linear Polynomial: Degree 1 (e.g., 3z + 7).
Constant Polynomial: Degree 0 (e.g., 8, written as 8x⁰).

2.1.5 Exercise Set 2.1

Content: Problems on finding degrees of polynomials, writing polynomials of specific degrees, and identifying coefficients and constant terms.

2.2 Linear Polynomials

2.2.1 Introduction to Linear Polynomials

Definition: Polynomials of degree 1 are called linear polynomials.
Focus of Chapter: Detailed study of linear polynomials.

2.2.2 Example: Perimeter of a Square

Perimeter: For a square of side x, the perimeter is 4x.
Nature: 4x is a linear polynomial in the variable x.
Observation: If side increases by 0.5 cm, perimeter increases by 2 cm (constant difference).

2.2.3 Example: Chess Club Charges

Cost Structure: ₹200 joining fee + ₹50 for every match played.
Total Cost Expression: ₹(200 + 50m), where m is the number of matches.
Nature: 200 + 50m is a linear polynomial in m.
Observation: Amount increases by a constant ₹50 for every additional match.

2.2.4 Linear Equations

Formation: Equating a linear polynomial in one variable to a constant value.

2.2.5 Example: Solving for Two Numbers

Problem: Sum of two numbers is 64; one is 10 more than the other.
Linear Equation: x + (x + 10) = 64, which simplifies to 2x + 10 = 64.
Solution: x = 27; the numbers are 27 and 37.

2.2.6 Polynomials as Input-Output Processes (Functions)

Concept: For every input value of the variable, a polynomial produces a corresponding output value.
Example: For 2x + 3, if x = 4, output is 11; if x = -6, output is -9.
Distinction: 2x + 3 is a linear function, 10x - x² is a quadratic function.

2.2.7 Exercise Set 2.2

Content: Problems on evaluating linear and quadratic polynomials, and solving word problems that translate into linear equations.

2.3 Exploring Linear Patterns

2.3.1 Growing Pattern of Square Tiles

Pattern: Each stage is obtained by adding two more tiles to the previous stage (1, 3, 5, 7...).
Generalisation: Number of squares at Stage n is 2n - 1.
Nature: 2n - 1 is a linear polynomial of degree 1.
Constant Difference: The difference between consecutive terms is constant (2).

2.3.2 Example: Bela's Pocket Money

Problem: Bela has ₹100, spends ₹5 every day.
Amount Left (nth day): ₹(100 - 5n).
Solution: ₹40 will be left on the 12th day.

2.3.3 Example: Auto-Rikshaw Fare

Fare Structure: ₹25 for initial 2 km, then increases by ₹15 per km thereafter.
Total Fare (n km, for n ≥ 2): ₹25 + 15(n - 2) = ₹(15n - 5).
Solution: Fare for 10 km is ₹145.

2.3.4 Definition of Linear Patterns

Characteristic: A sequence of numbers where the difference between two consecutive terms is constant.
Nature: The nth term in a linear pattern is a linear expression in n.

2.3.5 Exercise Set 2.3

Content: Solving problems involving savings, member dropouts, area of rectangles, volume of boxes, and pages left in a book, all expressed as linear patterns.

2.4 Linear Growth and Linear Decay

2.4.1 Introduction to Linear Growth and Decay

Concept: Linear expressions are used to model situations where there is a consistent increase (growth) or decline (decay).

2.4.2 Example: Linear Growth in Journey Cost

Function: C(d) = 100 + 60d, where C is cost in ₹, d is distance in km.
Observation: As d increases by 1 km, C increases by a fixed ₹60.
Nature: This is an example of linear growth.

2.4.3 Example: Linear Decay in Water Height

Function: h(t) = 3 - 0.5t, where h is height in m, t is months.
Observation: As t increases by one month, h decreases by a fixed 0.5 m.
Nature: This is an example of linear decay.

2.4.4 Formal Definitions

Linear Growth: A linear pattern where a quantity increases by a constant amount over equal intervals.
Linear Decay: A linear pattern where a quantity decreases by a constant amount over equal intervals.

2.4.5 Exercise Set 2.4

Content: Analyzing plant height, mobile phone depreciation, village population, and prepaid balance as examples of linear growth or decay.

2.5 Linear Relationships

2.5.1 General Form of Linear Relationships

Representation: A linear relationship between two variables x and y can be expressed as y = ax + b.
Example (Square Tiles Revisited): y = 2x - 1, where x is term number and y is number of square tiles.

2.5.2 Example: Finding 'a' and 'b' from Data Points

Problem: Telecom bill (y) depends on data used (x). Given (10 GB, ₹350) and (20 GB, ₹550).
Method: Substitute given points into y = ax + b to form a system of linear equations.
Solution: a = 20, b = 150.
Linear Relationship: y = 20x + 150 (₹20 per GB + ₹150 fixed monthly fee).

2.5.3 Exercise Set 2.5

Content: Problems on finding the values of 'a' and 'b' in linear relationships given two data points (e.g., learning platform, gym charges, temperature conversion).

2.6 Visualising Linear Relationships

2.6.1 Graphing Linear Equations

Method: To plot a linear equation (y = ax + b) as a straight line, identify any two points that satisfy the equation, plot them on a coordinate plane, and join them.
Verification: A point lies on a line if its coordinates satisfy the line's equation.

2.6.2 Example: Identifying an Equation from Plotted Points (`y = 3x`)

Points: (-1, -3), (0, 0), (1, 3), (3, 9), (4, 12).
Observation: The y-coordinate is consistently three times the x-coordinate.
Equation: y = 3x.

2.6.3 Example: Identifying an Equation from Plotted Points (`y = -2x`)

Points: (-3, 6), (-2, 4), (0, 0), (1, -2), (2, -4), (3, -6).
Observation: The y-coordinate is consistently negative two times the x-coordinate.
Equation: y = -2x.

2.6.4 Example: Graphs of `y = ax` (for `a > 0`)

Observation 1: Equations of the form y = ax always pass through the origin (0, 0).
Observation 2: If a > 1, the line is steeper than y = x.
Observation 3: If a < 1, the line is less steep than y = x.
Slope 'a': Represents the steepness of the line.

2.6.5 Example: Graphs of `y = -ax` (for `a > 0`)

Observation: These lines generally slant downwards from left to right, indicating a negative slope.
Effect of 'a': As 'a' increases (e.g., from 1/3 to 3), the line becomes steeper in the downward direction.

2.6.6 Example: Graphs of `y = ax + b` (varying `b`)

Observation: If 'a' (slope) is kept fixed while 'b' (y-intercept) varies (e.g., y = 2x - 1, y = 2x + 1, y = 2x + 5), the lines remain parallel to each other.
Effect of 'b': The lines shift vertically along the y-axis.

2.6.7 The Y-intercept

Definition: For an equation y = ax + b, 'b' is the y-intercept.
Location: The line cuts the y-axis at the point (0, b).
Interpretation: Distance from the origin where the line crosses the y-axis (positive or negative direction).

2.6.8 Key Conclusions on Linear Graphs

Equation y = ax + b: 'a' is the slope, 'b' is the y-intercept.
Changing 'a' (fixed 'b'): Slope changes, y-intercept remains fixed.
Changing 'b' (fixed 'a'): Lines shift vertically, remaining parallel to the original line.
Linear Growth: Represented by a straight line with a positive slope.
Linear Decay: Represented by a straight line with a negative slope.
Slope in Linear Patterns: Represents the constant difference between consecutive terms of a sequence.

2.6.9 Exercise Set 2.6

Content: Drawing graphs for various sets of lines and reflecting on the roles of 'a' (slope) and 'b' (y-intercept).

2.7 End-of-Chapter Exercises

Content: A comprehensive set of problems covering polynomial definition, evaluation, solving linear equations from word problems, graph plotting, identifying slopes and y-intercepts, and real-world applications of linear functions.

2.8 Chapter Summary

Algebraic Expression: Combines numbers, variables, and operation symbols; contains terms, coefficients, and constants.
Univariate Polynomials: Algebraic expressions in one variable; their degree is the highest power of that variable.
Linear Polynomial: A polynomial with a degree of one (e.g., 2x + 3).
Linear Growth: Quantity increases by a fixed amount over equal intervals.
Linear Decay: Quantity decreases by a fixed amount over equal intervals.
Linear Pattern: A sequence where the difference between consecutive terms is constant.
Linear Relationship (y = ax + b): Represented by a straight line; 'a' is the slope, 'b' is the y-intercept.
Line Through Origin: If b = 0, the equation is y = ax, and the line passes through (0, 0).
Growth/Decay on Graph: Linear growth has a positive slope; linear decay has a negative slope.
Parallel Lines: Lines with equal slopes ('a' is fixed) but different y-intercepts ('b' varies).