Class 9 Maths Chapter 2: Linear Polynomials – Formulas, Rules & Key Study Guide
1. Fundamentals of Polynomials
An algebraic expression is a mathematical combination of numbers, variables, and operation symbols (e.g., 2x² + 5xy – 3y²).
- Terms: The individual parts of the expression separated by plus or minus signs (e.g., 2x², 5xy, and -3y² are terms).
- Coefficients: The numbers multiplying the variables in each term (e.g., in the term 2x², 2 is the coefficient; in 5xy, 5 is the coefficient; and in -3y², -3 is the coefficient).
- Univariate Polynomials: Special algebraic expressions that involve only one variable and its non-negative integer powers (e.g., x² + 5x + 3).
- Degree: The highest power of the variable in a univariate polynomial.
2. Classification of Polynomials by Degree
Polynomials are classified based on their highest power (degree):
| Polynomial Type | Degree | General Form | Source Example |
|---|---|---|---|
| Constant Polynomial | 0 | k (where k is a constant number) | 8 (can be written as 8x⁰) |
| Linear Polynomial | 1 | ax + b (where a ≠ 0) | 3z + 7 or 2x + 3 |
| Quadratic Polynomial | 2 | ax² + bx + c (where a ≠ 0) | x² + 5x + 1 |
| Cubic Polynomial | 3 | ax³ + bx² + cx + d (where a ≠ 0) | 5y³ + y² + 2y – 1 |
3. The Core Formula: General Form of a Linear Relationship
A linear relationship represents the connection between two variables, x and y, and is mathematically expressed as:
y = ax + b
Where:
- x and y are the variables.
- a represents the slope (or the constant rate of change) of the line.
- b represents the y-intercept (the starting value).
4. Key Graphing Rules and Visualizations
When plotting linear equations on a coordinate plane, the following rules apply:
- The Straight Line Rule: Any linear relationship expressed in the form y = ax + b is represented visually as a straight line.
- The Y-Intercept Coordinate: A line written as y = ax + b will always cut the y-axis at the exact coordinate point (0, b). The length b is the distance from the origin where the line intersects the y-axis.
- Example 1: The line y = 2x + 5 cuts the y-axis at A(0, 5).
- Example 2: The line y = 3x – 2 cuts the y-axis at C(0, -2).
- Passing through the Origin: When the constant b is 0, the equation becomes y = ax. Any such straight line will always pass through the origin (0, 0).
- Steepness of the Line (y = ax where a > 0):
- If a > 1, the line is steeper than the standard diagonal line y = x.
- If a < 1, the line is less steep than the standard diagonal line y = x.
5. Slope, Growth, and Decay Rules
- Linear Pattern: A sequence of numbers where the difference between two consecutive terms is constant (e.g., 1, 3, 5, 7, 9…). The slope (a) of the line represents this constant difference.
- Linear Growth: A pattern where a quantity increases by a constant/fixed amount over equal intervals. Graphically, it is always represented by a straight line with a positive slope (where a > 0).
- Linear Decay: A pattern where a quantity decreases by a constant/fixed amount over equal intervals. Graphically, it is always represented by a straight line with a negative slope (where a < 0).
6. The Rule of Parallel Lines
- Parallel Lines Rule: Straight lines that share the same slope (a) but have different y-intercepts (b) are parallel to each other and will never intersect.
- They are represented by equations of the form y = ax + b, where a is fixed while b varies.
- Example: The lines y = 2x – 1, y = 2x + 1, and y = 2x + 5 all have a slope of 2 and are parallel to each other.