Introduction to Algebra

• Algebra uses symbols to represent numbers and quantities
• It is a generalization of arithmetic, where specific numerical values are used

Core Concepts

• Variables are symbols (usually letters) representing unknown or changing quantities
• Constants are fixed numerical values
• Expressions combine variables, constants, and operations
• Equations are statements asserting the equality of two expressions
• Polynomials consist of variables and coefficients, involving only addition, subtraction, and non-negative integer exponents

Basic Operations

• Addition (+): Combining terms
• Subtraction (-): Finding the difference between terms
• Multiplication (* or ×): Repeated addition of terms
• Division (/ or ÷): Splitting a term into equal parts
• Exponentiation (^): Raising a term to a power
• Root Extraction: Finding the root of a term

Laws of Algebra

• Commutative Law:
  • Addition: a + b = b + a
  • Multiplication: a * b = b * a
• Associative Law:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a * b) * c = a * (b * c)
• Distributive Law: a * (b + c) = a * b + a * c
• Identity Law:
  • Addition: a + 0 = a
  • Multiplication: a * 1 = a
• Inverse Law:
  • Addition: a + (-a) = 0
  • Multiplication: a * (1/a) = 1, where a ≠ 0

Solving Equations

• Linear Equations: Equations where the highest power of the variable is 1
  • General form: ax + b = 0
  • Solving involves isolating the variable using algebraic operations
• Quadratic Equations: Equations where the highest power of the variable is 2
  • General form: ax² + bx + c = 0
  • Solving methods:
    • Factoring
    • Completing the square
    • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
• Systems of Equations: A set of two or more equations with the same variables
  • Solving methods:
    • Substitution
    • Elimination
    • Graphing

Polynomials

• Polynomials consist of variables and coefficients
• Operations:
  • Addition: Combining like terms
  • Subtraction: Subtracting like terms
  • Multiplication: Distributing each term of one polynomial to each term of the other
  • Division: Polynomial long division or synthetic division
• Factoring: Expressing a polynomial as a product of simpler polynomials
  • Common techniques:
    • Factoring out the greatest common factor (GCF)
    • Difference of squares: a² - b² = (a + b)(a - b)
    • Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²
    • Grouping

Functions

• Definition: A relation between a set of inputs (domain) and a set of permissible outputs (range) where each input relates to exactly one output
• Notation: f(x), where x is the input and f(x) is the output
• Types of functions:
  • Linear functions: f(x) = mx + b
  • Quadratic functions: f(x) = ax² + bx + c
  • Polynomial functions
  • Rational functions: f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials
  • Exponential functions: f(x) = a^x
  • Logarithmic functions
  • Trigonometric functions
• Graphing functions: Visual representation of a function on a coordinate plane

Inequalities

• Definition: Mathematical statements comparing two expressions using inequality symbols
  • Symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to)
• Solving Inequalities is similar to solving equations, but with some differences:
  • Multiplying or dividing by a negative number reverses the inequality sign
• Interval Notation: Represents a set of numbers using intervals
  • Examples: (a, b), [a, b], (a, ∞), [-∞, b]
• Compound Inequalities: Two or more inequalities joined by "and" or "or"
  • Solving involves solving each inequality separately and then finding the intersection (for "and") or union (for "or") of the solution sets.

Radicals

• Definition: A radical expression consists of a radical symbol, a radicand, and an index
• Simplification: Reducing a radical expression to its simplest form
• Operations:
  • Addition and Subtraction: Combining like radicals
  • Multiplication: Multiplying the coefficients and radicands separately
  • Division: Rationalizing the denominator
• Rationalizing the Denominator: Eliminating radicals from the denominator of a fraction

Exponents

• Definition: An exponent indicates how many times a base is multiplied by itself
• Rules of exponents:
  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n)
  • Power of a power: (a^m)^n = a^(m*n)
  • Power of a product: (a * b)^n = a^n * b^n
  • Power of a quotient: (a / b)^n = a^n / b^n
  • Zero exponent: a^0 = 1 (where a ≠ 0)
  • Negative exponent: a^(-n) = 1 / a^n
• Fractional Exponents: Represent roots
  • a^(1/n) = nth root of a
  • a^(m/n) = (a^(1/n))^m = (a^m)^(1/n)

Logarithms

• Definition: The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number
• Notation: logₐ(x) = y, where a is the base, x is the argument, and y is the exponent
• Properties of Logarithms:
  • Product rule: logₐ(xy) = logₐ(x) + logₐ(y)
  • Quotient rule: logₐ(x/y) = logₐ(x) - logₐ(y)
  • Power rule: logₐ(x^p) = p * logₐ(x)
  • Change of base formula: logₓ(a) = logᵧ(a) / logᵧ(x)
• Common Logarithms: Base 10 logarithms (log₁₀(x) or simply log(x))
• Natural Logarithms: Base e logarithms (logₑ(x) or ln(x))
• Solving Logarithmic and Exponential Equations: Using the properties of logarithms and exponents to isolate the variable

Complex Numbers

• Definition: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)
• Operations:
  • Addition and Subtraction: Combining real and imaginary parts separately
  • Multiplication: Using the distributive property and the fact that i² = -1
  • Division: Multiplying the numerator and denominator by the conjugate of the denominator
• Complex Conjugate: The complex conjugate of a + bi is a - bi
• Absolute Value (Modulus): The distance from the origin to the point representing the complex number in the complex plane
• Representation: Complex numbers can be represented graphically on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.