Class 9 Maths Polynomials PPT (53 Slides) – Introduction to Linear Polynomials | USP Indore

Class 9 Maths Polynomials PPT—Introduction to Linear Polynomials | 53 Slides | USP Indore

Download the complete 53-slide PowerPoint presentation for Class 9 Maths—Polynomials: Introduction to Linear Polynomials, prepared by Unique Study Point (USP), Indore. Based on the NCERT Ganita Manjari syllabus (Session 2026-27), this PPT covers all 6 sections of the chapter with real-life examples, practice problems, NCERT exercises, and a full chapter summary.

Learning Journey — 6 Sections Across 53 Slides

📘 Section 2.1 — Algebraic Expressions and Polynomials (Slides 3–9)

• What is an algebraic expression? — terms, variables, coefficients, constants
• Example: 3x + 5y −—identifyingng coefficient, variable, and constant
• Polynomials in one variable (univariate polynomials) — definition and examples
• Degree of a polynomial — highest power of the variable
• Types of polynomials by degree: Constant (degree 0, e., 8);), Linear (degree 1e.g.,g. 3z 7);), Quadratic (degree 2e.g.,g. x² + 5x 1);), Cubic (degree 3e.g.,g. 5y³ + y² + 2y − 1)
• Real-life example — The Decorated Garden: Total cost = Rs (200ℓ + 160w + 50ℓw)—expression with TWO variables ℓ and w
• Difference between polynomial in one variable and expression in two variables

📗 Section 2.2 — Linear Polynomials (Slides 10–18)

• ⁣ hat makes a polynomial LINEAR—its degree must be exactly 1
• General form of a linear polynomial: ax + b (where a ≠ 0)
• Linear patter—constantnt difference between consecutive values
• Linear equations from linear polynomials — forming and solving
• A polynomial as an input-output machine (function): put in any x → get exactly one output—thisis is called a FUNCTION
• Example: y = 2x + 3; input x = 4 → output = 11; input x = −6 → output = −9
• Linear vs quadratic functio—comparisonon of graphs (straight line vs curve)
• Practice Set 2.2 Part 1 — Problems 1 to 4
• Practice Set 2.2 Part 2: Ruby\'s coins problem (3× two-rupee coins as five-rupee); 300 ft fence cut into two pieces (longer = 4× shorter); rectangle length = 2w + 3, perimeter = 24 —findnd dimensions

📙 Section 2.3 — Exploring Linear Patterns (Slides 19–24)

• ⁣ n a linear pattern, the difference between consecutive values is CONSTANT
• Number line patter—equalal steps of +50: 250, 300, 350, 400 → equal steps = a straight line!
• Square perimeters: 4 → 6 → 8 (each +2) — linear pattern
• Growing tile pattern and the rule 2n − 1 (Stage 1: 1 tile, Stage 2: 4 tiles, Stage 3: 7 tiles)
• nth term as a linear expression
• Real-life linear patterns: pocket money savings, auto fare (fixed charge + per km charge)
• Practice problems on identifying and forming linear pattern rules

📕 Section 2.4 — Linear Growth and Linear Decay (Slides 25–29)

• Linear GROW—a a quantity rises by a fixed amount each step (positive slope)
• Linear DEC—a a quantity falls by a fixed amount each step (negative slope)
• Real-life examples of linear growth: salary increments, savings, plant height
• Real-life examples of linear decay: candle burning, fuel consumption, depreciation
• Forming linear polynomial expressions for growth and decay situations
• Practice proble—identifyingng growth vs decay; writing the linear expression

📓 Section 2.5 — Linear Relationships (y = ax + b) (Slides 30–34)

• Standard form of a linear relationship: y = ax + b
• Finding values of a and b from given data points
• ⁣ f two data points are known, how do you find the unique linear polynomial that fits them
• ⁣ eal-life examples: taxi fare formula, temperature conversion, simple interest
• Practice proble—findingng a and b; writing the linear relationship equation

📒 Section 2.6 — Visualising Lines on a Graph (Slides 35–44)

• How to plot a line from a linear polynomi: — pick two points, join them
• Step-by-step example: y = 2x + 1; when = 0, 0 → A(0, 1); when = 3, 3 → B(3, 7); draw line through A and B
• A point lies on the line ONLY if its (x, y) fits the equation
• ⁣ hat the slope really tells us: slope = the constant DIFFERENCE between consecutive terms of a sequence
• Compare y = 3x + 1 and y = −3x +—onene rises (positivslope), and the), other falls (negative slope), yet BOTH cross y-axis at (0, 1)
• y-intercept = value of b in y = ax +, = where line crosses y-axis
• Effect of changing slope a and intercept b on the graph
• Practice proble—plottingng lines, identifying slope and y-intercept, checking whether a point lies on a line

Chapter Summary — Slides 45–46

• An algebraic expression combines numbers, variables and operations (terms, coefficients, constants)
• A univariate polynomial has ONE variable; its degree is the highest power
• Degree 1 = linear; Degree 2 = quadratic; Degree 3 = cubic; Degree 0 = constant
• A linear pattern has a CONSTANT difference between consecutive term(e.g.,g. 2, 5, 8, 11 → +3 each time)
• A linear polynomial used as a function: input x → output y = ax + b
• Graph of a linear polynomial = a straight line; slope = constant difference; y-intercept = value of b

NCERT Exercises — All 11 Questions Covered (Slides 47–52)

• Q1. Identify which expressions are polynomials and which are not. Give reasons.
• Q2. Write the degree of each given polynomial.
• Q3. Identify the type (linear/quadratic/cubic/constant) of each given polynomial.
• Q4. A sequence is given — determine if it is linear. If yes, write the linear polynomial for its nth term.
• Q5. A linear polynomial p(x) gives p(2) = 7 and p(−1) = 1. Find p(x).
• Q6. Ruby\'s taxi charges Rs 15 for the first km and Rs 8 for each subsequent km. Write a linear polynomial for the fare for n km and find the fare for 10 km.
• Q7. The temperature of a city starts at 5°C and rises by 3°C every hour. Write a linear polynomial for temperature after t hours. When will it reach 20°C?
• ⁣⁣Q8. Plot the graph of y = −2x + 4. Find where it crosses the x-axis and y-axis.
• Q9. Work = constant force × distance. Express as a linear equation in two variables (work w and distance d) with force = 3 units. Draw its graph. What is the work when the distance is 2 units?
• ⁣⁣Q10. The graph of a linear polynomial p(x) passes through (1, 5) and (3, 11). (i) Find p(x). (ii) Find where its graph cuts the axes. (iii) Draw and verify.
• Q11. Let p(x) = ax + b and q(x) = cx + d with p(0) 5,5; p(x) − q(x) cutthe x-axisis at (30),); and p(x) + q(x) = 6x + 4 for all x. Find p(x) and q(x).