Algebra Fundamentals

Study Notes

Algebra

Definition

Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through the use of symbols, equations, and functions.

Key Concepts

Variables: Letters or symbols that represent unknown values or quantities.
Constants: Numbers that do not change value.
Expressions: Combinations of variables, constants, and mathematical operations.
Equations: Statements that express the equality of two mathematical expressions.
Functions: Relations between variables, often expressed as output values dependent on input values.

Types of Equations

Linear Equations: Equations in which the highest power of the variable(s) is 1.
- Example: 2x + 3 = 5
Quadratic Equations: Equations in which the highest power of the variable(s) is 2.
- Example: x^2 + 4x + 4 = 0
Polynomial Equations: Equations consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- Example: x^3 - 2x^2 + x - 1 = 0

Algebraic Operations

Addition and Subtraction: Combining like terms by adding or subtracting coefficients.
Multiplication: Distributing numbers and variables to combine terms.
Division: Dividing both sides of an equation by a common factor.

Solving Equations

Linear Equations: Solve by adding, subtracting, multiplying, or dividing both sides by a value to isolate the variable.
Quadratic Equations: Solve using factoring, quadratic formula, or completing the square.

Applications of Algebra

Science and Engineering: Modeling real-world phenomena, such as motion, energy, and population growth.
Computer Science: Developing algorithms and data structures.
Cryptography: Creating secure encryption methods.

Description

Learn the basics of algebra, including variables, constants, equations, and functions. Explore types of equations, algebraic operations, and solving methods. Discover applications of algebra in science, computer science, and cryptography.