Introduction to Algebra

• Algebra uses symbols to represent numbers and quantities, generalizing arithmetic operations for abstract problem-solving

Variables and Constants

• Variables are typically letters representing unknown or changing amounts
• Constants are fixed, unchanging values

Expressions and Equations

• Algebraic expressions combine variables, constants, and operations like addition, subtraction, multiplication, division, and exponentiation
• Equations state the equality of two expressions using an equals sign (=)

Basic Operations

• Addition: Combines terms
• Subtraction: Finds the difference
• Multiplication: Finds the product
• Division: Finds the quotient
• Exponentiation: Raises a term to a power

Order of Operations

• Use PEMDAS/BODMAS for correct evaluation:
  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)

Terms and Coefficients

• A term is a single number, variable, or their product
• A coefficient is the numerical part multiplying the variable in a term

Like Terms

• Like terms have identical variables raised to the same powers
• Combine like terms by adding/subtracting coefficients

Simplifying Expressions

• Simplification combines like terms and performs operations to reach the simplest form

Distributive Property

• The distributive property: a(b + c) = ab + ac
• This allows multiplication of a term across parentheses

Solving Equations

• Solving identifies variable values that make the equation true
• Inverse operations isolate the variable

Linear Equations

• These equations have a maximum variable power of 1
• A linear equation's general form: ax + b = 0 (a, b are constants; x is the variable)

Solving Linear Equations

• Use inverse operations
• Adding/subtracting the same value from both sides maintains equality
• Multiplying/dividing by the same non-zero value maintains equality

Systems of Linear Equations

• A system is two or more linear equations with the same variables
• The solution satisfies all equations in the system

Methods for Solving Systems of Equations

• Substitution: Solve for one variable, then substitute
• Elimination: Add/subtract equations to eliminate a variable
• Graphing: Find intersection points of each equation's graph

Inequalities

• Inequalities compare expressions:
  • <: less than

  • : greater than

  • ≤: less than or equal to
  • ≥: greater than or equal to
  • ≠: not equal to

Solving Inequalities

• Similar to equations, but:
  • Multiplying/dividing by a negative number reverses the inequality sign

Graphing Inequalities

• Represent inequalities on a number line
• Open circle for < and >
• Closed circle for ≤ and ≥

Polynomials

• Polynomials use variables and coefficients with addition, subtraction, multiplication, and non-negative integer exponents
• Examples:
  • Monomial: Single term (e.g., 5x)
  • Binomial: Two terms (e.g., 3x + 2)
  • Trinomial: Three terms (e.g., x^2 + 2x + 1)

Operations with Polynomials

• Adding/Subtracting: Combine like terms
• Multiplying: Distribute
• Dividing: Use long or synthetic division

Factoring Polynomials

• Expressing a polynomial as a product of simpler factors
• Techniques:
  • Greatest Common Factor (GCF)
  • Difference of Squares: a^2 - b^2 = (a + b)(a - b)
  • Perfect Square Trinomials: a^2 + 2ab + b^2 = (a + b)^2
  • Quadratic Trinomials: Factoring ax^2 + bx + c

Quadratic Equations

• Equations with a maximum variable power of 2
• General form: ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0

Methods for Solving Quadratic Equations

• Factoring: Factor and set each factor to zero
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Completing the Square: Form a perfect square trinomial

Functions

• A relation where each input has exactly one output
• Inputs are the argument of the function and outputs are the value

Function Notation

• Functions are often denoted by f(x), where x is the input and f(x) is the output

Domain and Range

• Domain: All possible input (x) values
• Range: All possible output (f(x)) values

Linear Functions

• Expressed as f(x) = mx + b (m = slope, b = y-intercept)

Slope-Intercept Form

• y = mx + b (m = slope, b = y-intercept)

Slope

• Steepness and direction of a line
• Calculated as rise over run (change in y / change in x)

Exponential Functions

• f(x) = a^x (a is a constant base, x is the exponent)

Logarithmic Functions

• Inverses of exponential functions
• log_a(x) gives the exponent needed to raise 'a' to, to get 'x'