Introduction to Algebra: Concepts and Operations

• Algebra uses symbols to represent numbers and quantities
• It generalizes arithmetic operations and relationships
• It allows for the manipulation of variables
• It allows for the solving of equations

Basic Concepts

• Variable: A symbol represents an unknown or changeable quantity, often a letter
• Constant: A fixed, unchanging value
• Expression: Combination of variables, constants, and arithmetic operations
• Equation: Statement of equality between two expressions, using an equals sign (=)
• Coefficient: Number multiplied by a variable in an algebraic expression
• Term: Single number or variable, or numbers and variables multiplied

Operations

• Addition (+): Combining terms
• Subtraction (-): Finding the difference between terms
• Multiplication (* or ·): Scaling terms
• Division (/ or ÷): Splitting terms into equal parts
• Exponentiation (^): Raising a term to a power
• Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right))

Expressions

• Algebraic Expression: A phrase with numbers, variables (like x, y), and operators (+, -, *, /)
• Examples: 3x + 2y - 5, a^2 - b^2, (x + 1) / (y - 2)
• Simplifying: Combine like terms to reduce to simplest form
• Example: 2x + 3x - y + 4y simplifies to 5x + 3y

Equations

• Definition: Statement asserting the equality of two expressions
• General Form: Expression = Expression
• Examples: x + 5 = 10, 2y - 3 = 7, x^2 + 2x + 1 = 0
• Solving: Finding values of variables that make the equation true
• Basic Principles:
  • Addition/Subtraction Property of Equality: Adding/subtracting the same value from both sides maintains equality
  • Multiplication/Division Property of Equality: Multiplying/dividing both sides by the same non-zero value maintains equality

Linear Equations

• Definition: Equation where the highest variable power is 1
• General Form: ax + b = c, where a, b, c are constants and x is the variable
• Solving: Isolate the variable using algebraic operations
• Example:
  • Equation: 3x + 5 = 14
  • Subtract 5: 3x = 9
  • Divide by 3: x = 3

Systems of Linear Equations

• Definition: Set of two or more linear equations with the same variables
• General Form:
  • a1x + b1y = c1
  • a2x + b2y = c2
• Methods for Solving:
  • Substitution: Solve one equation for a variable, substitute into the other
  • Elimination: Add/subtract multiples of equations to eliminate a variable

Polynomials

• Definition: Expression with variables and coefficients, using addition, subtraction, multiplication, & non-negative exponents
• General Form: anx^n + an-1x^(n-1) + ... + a1x + a0, where an, an-1, ..., a1, a0 are coefficients and n is a non-negative integer
• Types:
  • Monomial: One term (e.g., 3x^2)
  • Binomial: Two terms (e.g., 2x + 1)
  • Trinomial: Three terms (e.g., x^2 - 3x + 4)
• Operations:
  • Addition/Subtraction: Combine like terms
  • Multiplication: Use the distributive property

Factoring Polynomials

• Definition: Expressing polynomial as a product of simpler factors
• Techniques:
  • Greatest Common Factor (GCF): Largest factor dividing all terms
  • Difference of Squares: a^2 - b^2 = (a + b)(a - b)
  • Perfect Square Trinomial: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2
  • Quadratic Trinomial: Factoring trinomials of the form ax^2 + bx + c
• Example:
  • Factor x^2 - 4:
    • Using difference of squares: x^2 - 4 = (x + 2)(x - 2)

Quadratic Equations

• Definition: Equation of the form ax^2 + bx + c = 0, where a, b, c are constants and a ≠ 0
• Methods for Solving:
  • Factoring: Factor the quadratic expression, set each factor to zero
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Discriminant: b^2 - 4ac (inside the square root of the quadratic formula); determines root nature:
  • If b^2 - 4ac > 0: Two distinct real roots
  • If b^2 - 4ac = 0: One real root (repeated root)
  • If b^2 - 4ac < 0: Two complex roots

Inequalities

• Definition: Statement comparing two expressions using <, >, ≤, ≥, ≠
• Solving: Similar to equations, but:
  • Multiplying/dividing by a negative number reverses the inequality sign
• Example:
  • Solve 2x + 3 < 7:
    • Subtract 3: 2x < 4
    • Divide by 2: x < 2

Functions

• Definition: Relation between inputs and outputs where each input has exactly one output
• Notation: f(x) = output of function f for input x
• Types:
  • Linear Functions: f(x) = mx + b, where m is slope, b is y-intercept
  • Quadratic Functions: f(x) = ax^2 + bx + c
  • Polynomial Functions: f(x) = anx^n + an-1x^(n-1) + ... + a1x + a0
• Domain: Set of all possible input (x) values for which the function is defined
• Range: Set of all possible output (y) values the function can produce