Algebra Class Basics

Algebra Study Notes

Basic Concepts

• Variables: Symbols (usually letters) used to represent unknown values.
• Constants: Fixed values that do not change.
• Expressions: Combinations of variables, constants, and operators (e.g., (2x + 3)).
• Equations: Statements that two expressions are equal (e.g., (2x + 3 = 7)).

Types of Algebra

1. Elementary Algebra

   • Focuses on basic operations and solving simple equations.
   • Includes concepts like integers, fractions, and basic polynomials.

2. Abstract Algebra

   • Studies algebraic structures such as groups, rings, and fields.
   • Involves more theoretical concepts than elementary algebra.

3. Linear Algebra

   • Concerns vector spaces and linear mappings between them.
   • Key concepts include matrices, determinants, and eigenvalues.

Key Operations

• Addition and Subtraction: Combining like terms (e.g., (3x + 5x = 8x)).
• Multiplication: Distributive property (e.g., (a(b + c) = ab + ac)).
• Division: Involves finding the inverse operation, important for solving equations.

Solving Equations

• One-variable equations: Isolate the variable on one side (e.g., (2x + 3 = 7) → (2x = 4) → (x = 2)).
• Linear equations: Form (ax + b = 0), solved by isolating (x).
• Quadratic equations: Form (ax^2 + bx + c = 0), solved using:
  • Factoring
  • Quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

Functions

• Definition: A relation that assigns exactly one output for each input.
• Notation: (f(x)) indicates a function (f) evaluated at (x).
• Types of Functions:
  • Linear functions: Form (f(x) = mx + b).
  • Quadratic functions: Form (f(x) = ax^2 + bx + c).

Graphing

• Coordinate System: Consists of an x-axis and a y-axis.
• Slope: Measure of the steepness of a line, calculated as (m = \frac{y_2 - y_1}{x_2 - x_1}).
• Intercepts: Points where the graph intersects the axes (x-intercept, y-intercept).

Factoring

• Factoring Techniques:
  • Common factors
  • Difference of squares: (a^2 - b^2 = (a + b)(a - b))
  • Trinomials: (ax^2 + bx + c) can be factored into ((px + q)(rx + s)).

Exponents and Radicals

• Laws of Exponents:
  • (a^m \cdot a^n = a^{m+n})
  • (\frac{a^m}{a^n} = a^{m-n})
  • ((a^m)^n = a^{mn})
• Radicals:
  • Simplifying roots (e.g., (\sqrt{a^2} = a)).
  • Rationalizing denominators.

Inequalities

• Definition: A statement that compares two expressions (e.g., (x > 5)).
• Solving Inequalities:
  • Similar steps to equations, but reverse the inequality when multiplying/dividing by a negative number.

Applications

• Used in fields such as physics, economics, engineering, and computer science.
• Real-world problem solving, modeling relationships, and analyzing trends.

Basic Concepts

• Variables represent unknown values, often denoted by letters.
• Constants are fixed values that do not change.
• Expressions combine variables, constants, and operators (e.g., (2x + 3)).
• Equations state that two expressions are equal (e.g., (2x + 3 = 7)).

Types of Algebra

• Elementary Algebra: Involves basic operations with integers, fractions, and simple equations.
• Abstract Algebra: Examines algebraic structures such as groups, rings, and fields with a greater theoretical focus.
• Linear Algebra: Deals with vector spaces and linear mappings, emphasizing matrices, determinants, and eigenvalues.

Key Operations

• Addition and Subtraction involve combining like terms (e.g., (3x + 5x = 8x)).
• Multiplication applies the distributive property (e.g., (a(b + c) = ab + ac)).
• Division is finding the inverse operation, crucial for solving equations.

Solving Equations

• Isolate the variable in one-variable equations (e.g., (2x + 3 = 7) simplifies to (x = 2)).
• Linear equations take the form (ax + b = 0) and are solved by isolating (x).
• Quadratic equations, presented as (ax^2 + bx + c = 0), can be solved through:
  • Factoring
  • Using the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

Functions

• Functions relate inputs to exactly one output and are often denoted as (f(x)).
• Linear Functions are represented as (f(x) = mx + b).
• Quadratic Functions are expressed as (f(x) = ax^2 + bx + c).

Graphing

• The coordinate system consists of an x-axis (horizontal) and a y-axis (vertical).
• Slope measures line steepness, calculated as (m = \frac{y_2 - y_1}{x_2 - x_1}).
• Intercepts are points where the graph crosses the axes, including x-intercept and y-intercept.

Factoring

• Factoring techniques include:
  • Identifying common factors.
  • Applying the difference of squares formula: (a^2 - b^2 = (a + b)(a - b)).
  • Factoring trinomials in the form (ax^2 + bx + c) into ((px + q)(rx + s)).

Exponents and Radicals

• Laws of Exponents include:
  • (a^m \cdot a^n = a^{m+n})
  • (\frac{a^m}{a^n} = a^{m-n})
  • ((a^m)^n = a^{mn})
• Radicals involve simplifying roots (e.g., (\sqrt{a^2} = a)) and rationalizing denominators.

Inequalities

• Inequalities compare two expressions, indicated with symbols like (>) (e.g., (x > 5)).
• Solving inequalities follows similar steps to equations but requires reversing the inequality when multiplying or dividing by a negative number.

Applications

• Algebra is applied in various fields, including physics, economics, engineering, and computer science.
• It aids in real-world problem solving, modeling relationships, and analyzing trends.