Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• These symbols usually represent quantities without fixed values, known as variables.
• Algebra is a broad part of mathematics, along with number theory, geometry and analysis.

Core Concepts

• Variables are symbols (usually letters) that represent unknown or changeable values.
• Constants are values that do not change.
• Expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation).
• Equations are statements that two expressions are equal, indicated by an equals sign (=).

Basic Operations

• Addition: Combining terms; like terms can be added together.
• Subtraction: Finding the difference between terms; like terms can be subtracted.
• Multiplication: Multiplying terms; coefficients and variables are multiplied separately.
• Division: Dividing terms; coefficients and variables are divided separately.
• Exponentiation: Raising a term to a power.

Order of Operations

• Parentheses (or brackets) first
• Exponents next
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)
• The mnemonic PEMDAS (Please Excuse My Dear Aunt Sally) is often used to remember the order.
• Sometimes known as BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Solving Equations

• The goal is to isolate the variable on one side of the equation.
• Use inverse operations to "undo" operations applied to the variable.
• Maintain balance by performing the same operation on both sides of the equation.

Linear Equations

• Equations where the highest power of the variable is 1.
• General form: ax + b = c, where a, b, and c are constants, and x is the variable.
• Solving involves isolating x by subtracting/adding constants and then dividing by the coefficient of x.

Quadratic Equations

• Equations where the highest power of the variable is 2.
• General form: ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Methods for solving include factoring, completing the square, and using the quadratic formula.
• Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
• Discriminant (b² - 4ac) determines the nature of the roots (solutions):
  • Positive discriminant: two distinct real roots.
  • Zero discriminant: one real root (a repeated root).
  • Negative discriminant: two complex roots.

Factoring

• Breaking down an expression into a product of its factors.
• Useful for solving quadratic equations and simplifying expressions.
• Common techniques include factoring out the greatest common factor (GCF), difference of squares, and trinomial factoring.

Systems of Equations

• A set of two or more equations with the same variables.
• Solutions are values for the variables that satisfy all equations simultaneously.
• Methods for solving include substitution, elimination (addition/subtraction), and graphing.

Inequalities

• Similar to equations, but use inequality symbols (>, <, ≥, ≤) instead of an equals sign.
• Solutions are ranges of values that satisfy the inequality.
• When multiplying or dividing by a negative number, the direction of the inequality sign must be reversed.

Functions

• A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Represented as f(x), where x is the input and f(x) is the output.
• Types of functions include linear, quadratic, polynomial, exponential, and logarithmic functions.

Polynomials

• Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Examples: x² + 3x - 4, 5x³ - 2x + 1
• Degree of a polynomial is the highest power of the variable.

Exponents and Radicals

• Exponents indicate the power to which a base is raised (e.g., x^n).
• Radicals (roots) are the inverse operation of exponentiation (e.g., √x).
• Rules of exponents:
  • x^m * x^n = x^(m+n)
  • x^m / x^n = x^(m-n)
  • (x^m)^n = x^(m*n)
  • x^0 = 1
  • x^(-n) = 1/x^n
• Rational exponents: x^(m/n) = nth root of x^m

Logarithms

• The inverse function of exponentiation.
• log_b(x) = y if and only if b^y = x, where b is the base.
• Properties of logarithms:
  • log_b(xy) = log_b(x) + log_b(y)
  • log_b(x/y) = log_b(x) - log_b(y)
  • log_b(x^n) = n * log_b(x)
• Common logarithm: base 10 (log_10(x) or log(x))
• Natural logarithm: base e (log_e(x) or ln(x)), where e ≈ 2.71828

Complex Numbers

• Numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).
• Operations on complex numbers: addition, subtraction, multiplication, and division.
• Complex conjugate of a + bi is a - bi.

Matrices

• Rectangular arrays of numbers arranged in rows and columns.
• Used to solve systems of equations, represent linear transformations, and in various areas of mathematics and computer science.
• Operations on matrices: addition, subtraction, multiplication, and scalar multiplication.
• Determinant of a square matrix is a scalar value that can be computed from its elements.

Sequences and Series

• A sequence is an ordered list of numbers.
• A series is the sum of the terms in a sequence.
• Arithmetic sequences have a constant difference between consecutive terms.
• Geometric sequences have a constant ratio between consecutive terms.

Binomial Theorem

• A formula for expanding expressions of the form (a + b)^n, where n is a positive integer.
• (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n, and (n choose k) is the binomial coefficient.

Absolute Value

• The distance of a number from zero on the number line.
• Denoted as |x|.
• |x| = x if x ≥ 0, and |x| = -x if x < 0.

Graphing

• Visual representation of algebraic relationships on a coordinate plane.
• Linear equations produce straight lines.
• Quadratic equations produce parabolas.
• Functions can be graphed to analyze their behavior (e.g., intercepts, slope, maximum/minimum values).

Mathematical Modeling

• Using algebraic equations and functions to represent real-world phenomena.
• Creating a mathematical representation of a problem to analyze and solve it.
• Involves defining variables, formulating equations, and interpreting the results.