REVISION SUMMARY: Introduction to Linear Polynomials (CBSE Class 9)
1. Chapter at a glance
- Algebraic expressions consist of terms, variables (letter-numbers), coefficients and constants.
- Polynomials are univariate (one-variable) algebraic expressions involving a variable and its powers; the highest power is the degree.
- Polynomials are classified by degree: degree 0 (constant), degree 1 (linear), degree 2 (quadratic), degree 3 (cubic).
- Linear polynomials (degree 1) produce linear patterns where the difference between successive values at equal intervals is constant.
- Equating a linear polynomial to a constant yields a linear equation.
- Linear growth occurs when a quantity increases by a fixed amount over equal intervals; linear decay occurs when it decreases by a fixed amount.
- A linear relationship between variables x and y is expressed as y = ax + b, where a is the slope and b is the y-intercept.
- Graphs of y = ax + b are straight lines; lines with the same a but different b are parallel; positive slope represents growth, negative slope represents decay.
2. Definitions, theorems and results
- An algebraic expression combines numbers, variables and operation symbols. Its parts are called terms; the numbers multiplying the variables are coefficients; a number without a variable is a constant. (No proof required.)
- A univariate (one-variable) polynomial is an algebraic expression involving only one variable and its powers. (No proof required.)
- The degree of a polynomial is the highest power of the variable in it. (No proof required.)
- Degree 0: constant polynomial (e.g., 8).
- Degree 1: linear polynomial (e.g., 3z + 7).
- Degree 2: quadratic polynomial (e.g., x² + 5x + 1).
- Degree 3: cubic polynomial (e.g., 5y³ + y² + 2y – 1).
- A linear pattern is a sequence of numbers where the difference between two consecutive terms is constant. (No proof required.)
- Linear growth: a quantity increases by a constant amount over equal intervals. (No proof required.)
- Linear decay: a quantity decreases by a constant amount over equal intervals. (No proof required.)
- A linear relationship between two variables x and y is expressed by the equation y = ax + b. (No proof required.)
- In y = ax + b, a is the slope of the line and b is the y-intercept (the point where the line cuts the y-axis is (0, b)). (No proof required.)
- Lines with equal slopes but different y-intercepts are parallel to each other. (No proof required.)
- Linear growth is represented by a straight line with positive slope; linear decay is represented by a straight line with negative slope. (No proof required.)
- No theorems, lemmas or axioms requiring proof appear in the chapter.
3. Formula sheet
| Formula / Expression |
Meaning of symbols |
| y = ax + b |
Linear relationship; a = slope, b = y-intercept |
| Perimeter of square = 4x |
x = side length |
| Area = x(10 – x) or 10x – x² |
x = length (width = 10 – x) |
| Cost = 200 + 50m |
m = number of matches |
| Amount left = 100 – 5n |
n = number of days |
| Fare = 15n – 5 (n ≥ 2) |
n = km travelled (after first 2 km) |
| Height h(t) = 3 – 0.5t |
t = months; linear decay |
| Cost C(d) = 100 + 60d |
d = distance (km); linear growth |
4. Solved-example patterns
- Evaluate a linear (or other) polynomial at given value(s) of the variable: Substitute the given value(s) directly into the expression and simplify.
- Identify degree, coefficients and constant term: Locate the highest power (degree); read the numerical multiplier of each power and the term without a variable.
- Form a linear expression or equation from a word problem (perimeter, cost, age, coins, etc.): Assign variable(s) to unknown quantity(ies), translate the given conditions into an algebraic expression/equation, then solve if required.
- Find the general term of a linear pattern from a table or description: Observe the constant difference; express the nth term as a linear polynomial (usually of form an + b or an – b).
- Determine a and b in y = ax + b given two points/conditions: Substitute each pair (x, y) to obtain two equations; solve the simultaneous equations for a and b.
- Draw the graph of y = ax + b or identify slope/y-intercept: Choose two convenient values of x, find corresponding y-values, plot the points and join with a straight line; read a as slope and b as y-intercept from the equation.
- Distinguish linear growth vs. decay or identify parallel lines: Compare signs of slopes (positive = growth, negative = decay); check whether slopes are identical.
5. Common mistakes and exam pitfalls
- Sign errors when forming expressions such as (10 – x) or when subtracting terms.
- Forgetting the condition n ≥ 2 (or similar) when writing piecewise linear expressions (e.g., fare after initial distance).
- Confusing degree with the number of terms or with the coefficient of the highest power.
- Omitting units (cm, ₹, etc.) in final answers of word problems.
- Assuming every pattern is linear without verifying constant difference.
- Plotting graphs without extending the line or without labelling axes/points correctly.
- Mixing up slope (a) and y-intercept (b) when asked to identify them from an equation or graph.
- Neglecting to verify that a point satisfies the equation before claiming it lies on the line.