Chapter 2 Notes: Introduction to Linear Polynomials

Algebraic expressions combine numbers, variables (letter-numbers), and operation symbols. In an expression like 4x + 5y + 3, the terms are 4x, 5y, and 3. The numbers 4 and 5 are coefficients of x and y respectively, while 3 is a constant. Understanding these components helps us model real-world situations mathematically.

A polynomial is a univariate algebraic expression in one variable with its powers. The degree of a polynomial is the highest power of the variable. Linear polynomials have degree 1. For example, 2x + 3 and 5 - 4y are linear polynomials. When equating a linear polynomial to a constant, we get a linear equation. Linear polynomials can be viewed as input-output processes where each input value produces a specific output.

A linear pattern is a sequence where the difference between consecutive terms is constant. For example, in a growing tile pattern (1, 3, 5, 7, 9...), each term increases by 2. The general term can be expressed as a linear polynomial. The expression 2n - 1 represents the number of tiles at stage n. Linear patterns show constant rates of change, which is a defining characteristic of linear relationships.

Linear growth occurs when a quantity increases by a fixed amount over equal intervals. For example, a plant growing 0.5 feet each month shows linear growth. Linear decay occurs when a quantity decreases by a fixed amount. A water tank losing water at a constant rate demonstrates linear decay. Both are modeled using linear functions of the form f(x) = ax + b, where the sign of 'a' determines growth (positive) or decay (negative).

A linear relationship between two variables x and y is expressed as y = ax + b. To find this relationship when given two data points, substitute both points into the equation and solve the system to find values of a and b. For instance, with points (x₁, y₁) and (x₂, y₂), we can determine the linear relationship that connects dependent variable y to independent variable x.

Linear equations y = ax + b can be graphed as straight lines. The slope 'a' determines the steepness and direction of the line. When a > 0, the line slopes upward (growth); when a < 0, it slopes downward (decay). The y-intercept 'b' shows where the line crosses the y-axis at point (0, b). Lines with the same slope but different y-intercepts are parallel. Points on the line satisfy its equation.

Understanding terminology is essential for working with linear polynomials. Coefficient: the numerical factor of a term. Variable: a letter representing an unknown value. Degree: the highest power of the variable. Univariate: having one variable. Slope: the rate of change in a linear relationship. Y-intercept: where a line crosses the y-axis. Parallel lines: lines with same slope but different y-intercepts.