Algebra 10. bekkur

Algebra

Definition
- Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts
- Variables: Symbols that represent numbers (e.g., x, y).
- Constants: Fixed values (e.g., 5, -3).
- Expressions: Combinations of variables, constants, and operations (e.g., 3x + 2).
- Equations: Mathematical statements that two expressions are equal (e.g., 2x + 3 = 7).
Basic Operations
- Addition, subtraction, multiplication, division of algebraic expressions.
- Order of Operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – often abbreviated as PEMDAS.
Types of Algebra
- Elementary Algebra: Focuses on basic operations and the simplest equations.
- Abstract Algebra: Deals with algebraic structures such as groups, rings, and fields.
Solving Equations
- Isolate the variable on one side of the equation.
- Use algebraic operations and properties (e.g., distributive property, inverse operations).
- Example: To solve 2x + 3 = 7, first subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to find x = 2.
Functions
- A relation between a set of inputs and a set of possible outputs.
- Common types include linear (y = mx + b), quadratic (y = ax² + bx + c), and exponential (y = a * b^x).
Graphing
- Represents equations visually using a coordinate system (x-y plane).
- Important graphs: lines for linear equations, parabolas for quadratic equations.
Factoring
- The process of breaking down expressions into simpler components (e.g., factoring x² - 5x + 6 into (x - 2)(x - 3)).
- Key techniques: Common factor extraction, grouping, and using special products (e.g., difference of squares).
Inequalities
- Mathematical statements indicating one expression is greater or less than another (e.g., 2x + 3 < 7).
- Use similar methods to equations but remember to reverse the inequality sign when multiplying/dividing by a negative number.
Polynomials
- An algebraic expression made up of variables and coefficients (e.g., 4x³ + 3x² - 2).
- Operations: Addition, subtraction, multiplication, division (through synthetic division or long division).
Applications
- Algebra is used in various fields such as physics, engineering, economics, and everyday problem-solving scenarios, such as budgeting and planning.

Algebra: Definition and Key Concepts

Algebra is a branch of mathematics focused on using symbols and the rules governing them.
Variables are symbols representing unknown numbers.
Constants are fixed values.
Expressions are combinations of variables, constants, and operations.
Equations state that two expressions are equal.

Basic Operations and Order of Operations

Basic operations include addition, subtraction, multiplication, and division.
The order of operations dictates the order in which operations are performed.
PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is a helpful acronym to remember the order of operations.

Types of Algebra

Elementary Algebra focuses on basic operations, simple equations, and properties.
Abstract Algebra explores algebraic structures such as groups, rings, and fields.

Solving Equations

The goal is to isolate the variable on one side of the equation using algebraic operations and properties.
Examples of properties include the distributive property and inverse operations.
For example, to solve 2x + 3 = 7, first subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to find x = 2.

Functions

Functions are relationships between input values (domain) and output values (range).
Examples of common functions include linear (y = mx + b), quadratic (y = ax² + bx + c), and exponential (y = a * b^x) functions.

Graphing

Graphs represent equations visually in a coordinate system (x-y plane).
Linear equations are represented by straight lines.
Quadratic equations are represented by parabolas.

Factoring

Factoring breaks down expressions into simpler components.
Techniques include common factor extraction, grouping, and special products like the difference of squares.
For example, x² - 5x + 6 can be factored into (x - 2)(x - 3).

Inequalities

Inequalities express one expression as greater or less than another.
Solving inequalities involves similar methods as solving equations, remembering to reverse the inequality sign when multiplying or dividing by a negative number.

Polynomials

Polynomials are algebraic expressions with variables and coefficients.
Common polynomial operations include addition, subtraction, multiplication, and division (using synthetic division or long division).