Algebra Fundamental Concepts and Operations

Study Notes

Fundamental Concepts

Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in mathematical expressions and equations.
It deals with relationships between variables and constants.
Variables are symbols (like x, y, or z) that represent unknown values.
Constants are fixed values (like 2, 5, or π).
Expressions are combinations of variables, constants, and mathematical operations (like addition, subtraction, multiplication, division, etc.).
Equations are mathematical statements that show the equality of two expressions.
Solving equations involves finding the value of the variable that makes the equation true.

Basic Operations

Addition: Combining quantities.
Subtraction: Finding the difference between quantities.
Multiplication: Combining equal quantities.
Division: Separating a quantity into equal parts.
Order of Operations (PEMDAS/BODMAS): A set of rules to determine the sequence of operations in an expression: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).
Properties of Operations: Commutative (order doesn't matter for addition and multiplication), associative (grouping doesn't matter for addition and multiplication), distributive (multiplication distributes over addition).

Types of Equations

Linear Equations: Equations that can be written in the form ax + b = 0, where 'a' and 'b' are constants and 'x' is the variable. Graphically, a linear equation represents a straight line.
Quadratic Equations: Equations that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'x' is the variable. Graphically, a quadratic equation represents a parabola.
Systems of Equations: Two or more equations with the same variables. Solving a system of equations means finding the values of the variables that satisfy all the equations in the system.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
Terms in a polynomial are separated by addition or subtraction signs.
The degree of a polynomial is the highest power of the variable in the polynomial.
Polynomials can be added, subtracted, multiplied, and divided.

Factoring

Factoring is the process of writing a polynomial as a product of other polynomials.
Common factors are identified and removed from terms.
Factoring is used to simplify expressions, solve equations, and perform calculations with polynomials.

Functions

A function is a relationship between input (often called the independent variable) values and output (often called the dependent variable) values where each input has only one output.
Functions can be represented algebraically, graphically, or numerically.
The domain of a function is the set of all possible input values.
The range of a function is the set of all possible output values.
Key functions include linear, quadratic, and exponential functions.

Solving Equations

Procedures for solving various types of equations are developed for different cases.
Methods for solving equations might involve isolating the variable, factoring, using the quadratic formula, or graphic representation.

Graphs and Representations

Graphs of equations visualize relationships between variables and show trends in the data (e.g. straight line for linear equation, parabola for quadratic).
Coordinate systems are used to represent solutions graphically.
Interpreting and analysing graphs is crucial to understanding the behavior of functions and equations.

Description

This quiz covers fundamental concepts and basic operations of algebra, including variables, constants, expressions, equations, and the order of operations. Test your understanding of how to manipulate and solve algebraic expressions and equations effectively.