Algebra Concepts and Equations

Algebra

• Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols.

• Key Concepts:

  • Variables: Symbols that represent unknown values (e.g., x, y).
  • Expressions: Combinations of numbers, variables, and operations (e.g., 2x + 3).
  • Equations: Mathematical statements asserting the equality of two expressions (e.g., 2x + 3 = 7).

• Types of Equations:

  • Linear Equations: Form ax + b = 0; graphs as straight lines.
  • Quadratic Equations: Form ax² + bx + c = 0; graphs as parabolas.
  • Polynomial Equations: Involves terms with variables raised to whole-number powers.

• Operations:

  • Addition and Subtraction: Combining like terms.
  • Multiplication: Distributive property (a(b + c) = ab + ac).
  • Factoring: Expressing an expression as a product of its factors (e.g., x² - 9 = (x - 3)(x + 3)).

• Functions:

  • Definition: A relation that assigns exactly one output for each input.
  • Notation: f(x) represents the function with respect to variable x.
  • Types: Linear, quadratic, polynomial, exponential, logarithmic.

Permutations

• Definition: Arrangements of objects in a specific order.

• Key Formula:

  • Permutations of n objects: n! (n factorial), where n! = n × (n - 1) × ... × 1.
  • Permutations of n objects taken r at a time: P(n, r) = n! / (n - r)!

• Key Concepts:

  • Distinct Objects: All objects are different; the order matters.
  • Identical Objects: Some objects are the same; adjust formula to account for duplicates:
    • P(n; n₁, n₂, ..., nₖ) = n! / (n₁! × n₂! × ... × nₖ!)

• Applications:

  • Combinatorial Problems: Arranging people, letters, or numbers in specific orders.
  • Probability: Calculating the likelihood of various outcomes based on arrangements.

• Example Problems:

  • Arranging 3 books on a shelf: 3! = 6 arrangements.
  • Choosing and arranging 2 from a set of 5 different fruits: P(5, 2) = 5! / (5 - 2)! = 20.

Algebra

• Algebra is a mathematical discipline focused on symbols and their manipulation according to rules.
• Variables are symbols (like x and y) denoting unknown values.
• Expressions combine numbers, variables, and operations (example: 2x + 3).
• Equations assert the equality between two expressions (example: 2x + 3 = 7).
• Linear Equations follow the format ax + b = 0 and produce straight-line graphs.
• Quadratic Equations have the form ax² + bx + c = 0, resulting in parabolic graphs.
• Polynomial Equations consist of terms with variables raised to non-negative integer powers.
• Operations in algebra include:
  • Addition/Subtraction: Focus on combining like terms for simplification.
  • Multiplication: Utilize the distributive property; for example, a(b + c) = ab + ac.
  • Factoring: The process of expressing an algebraic expression as a product of its factors (example: x² - 9 = (x - 3)(x + 3)).
• A Function relates each input to a single output, denoted as f(x) for variable x.
• Functions can be categorized as linear, quadratic, polynomial, exponential, or logarithmic.

Permutations

• Permutations involve arranging objects in a designated order.
• The permutations of n objects is calculated as n! (n factorial), which is n × (n - 1) ×...× 1.
• The formula for permutations of n objects taken r at a time is P(n, r) = n! / (n - r)!.
• Key Concepts:
  • Distinct Objects: Each object is unique, and the order of arrangement matters.
  • Identical Objects: When objects are similar, the formula adjusts for duplicates: P(n; n₁, n₂,..., nₖ) = n! / (n₁! × n₂! ×...× nₖ!).
• Applications:
  • Solve combinatorial problems by arranging items like people, letters, or numbers.
  • Aid in probability calculations based on the arrangements of different items.
• Example Problems:
  • Arranging 3 books yields 3! = 6 possible arrangements.
  • Selecting and arranging 2 fruits from a collection of 5 gives P(5, 2) = 5! / (5 - 2)! = 20 different combinations.