Introduction to Algebra

• Algebra involves symbols and rules to manipulate them.
• Symbols represent quantities without fixed values, known as variables.
• Algebra is a unifying branch of nearly all mathematics.

Basic Operations

• Addition, subtraction, multiplication, and division are the fundamental operations.
• These operations apply to numerical values and variables.

Variables and Constants

• Variables are symbols (usually letters) representing unknown or changeable values.
• Constants are fixed numerical values that remain unchanged.
• Expressions are combinations of variables, constants, and operations.
• Equations declare the equality of two expressions.

Algebraic Expressions

• Algebraic expressions are mathematical phrases containing numbers, variables, and operation symbols.
• Examples of algebraic expressions include 3x + 2y, a - 5, and x^2 + 4x - 7.
• Terms are parts of an algebraic expression separated by addition or subtraction; in "3x + 2y", "3x" and "2y" are terms.
• Coefficients are the numerical part of a term containing a variable; in "3x", 3 is the coefficient.
• Like terms share the same variable raised to the same power and can be combined (e.g., 3x + 5x = 8x).
• Unlike terms have different variables or the same variables raised to different powers and cannot be combined directly.

Evaluating Expressions

• Evaluating an expression means finding its numerical value by substituting specific values for the variables.
• The order of operations (PEMDAS/BODMAS) is vital when evaluating expressions.

Equations and Solving Equations

• Equations state the equality of two expressions, connected by an equals sign (=).
• Solving an equation means finding the value(s) of the variable(s) that satisfy the equation.
• A solution is the value(s) that satisfy the equation.
• Linear equations contain variables raised to the first power only.
• Quadratic equations contain variables raised to the second power.

Techniques for Solving Equations

• Isolation involves isolating the variable on one side of the equation by performing the same operations on both sides.
• Inverse operations use opposite operations to undo operations affecting the variable (e.g., using subtraction to undo addition).
• Combining like terms simplifies each side of the equation.
• Distributive property: a(b + c) = ab + ac; apply it to remove parentheses.
• Factoring expresses a quadratic expression as the product of two linear expressions and is used to solve quadratic equations.
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a); used to solve quadratic equations of the form ax^2 + bx + c = 0.

Linear Equations

• A linear equation can be written in the form ax + b = 0, where a and b are constants, and x is the variable.
• Solving linear equations involves isolating the variable on one side of the equation.
• Examples of linear equations include 2x + 3 = 7 and 5x - 1 = 9.
• Use inverse operations to isolate the variable when solving.

Systems of Linear Equations

• A system of linear equations features two or more linear equations with the same variables.
• Solving a system means finding values for the variables that satisfy all equations simultaneously.
• Methods for solving systems of linear equations:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: Add or subtract multiples of the equations to eliminate one of the variables.
  • Graphing: Graph each equation and find the point(s) of intersection, representing the solution(s).

Inequalities

• Inequalities compare two expressions using symbols like <, >, ≤, and ≥.
• Solving inequalities involves finding the range of values for the variable that make the inequality true.
• Operations are similar to solving equations, but multiplying or dividing by a negative number reverses the inequality sign.

Functions

• A function is a relationship between a set of inputs (domain) and a set of possible outputs (range), where each input is related to exactly one output.
• f(x) represents the output of the function f for an input x.
• Types of functions include:
  • Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic Functions: f(x) = ax^2 + bx + c.
  • Polynomial Functions: Functions involving non-negative integer powers of x.
  • Exponential Functions: f(x) = a^x, where a is a constant.
  • Logarithmic Functions: Inverse of exponential functions.
• Domain: Set of all possible input values (x-values) for a function.
• Range: Set of all possible output values (y-values) for a function.

Polynomials

• Polynomials are expressions with variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
• Standard Form: Polynomials are often written in descending order of exponents.
• Degree is the highest power of the variable.
• Types of Polynomials:
  • Monomial: A polynomial with one term (e.g., 5x^2).
  • Binomial: A polynomial with two terms (e.g., 3x + 2).
  • Trinomial: A polynomial with three terms (e.g., x^2 + 4x - 7).
• Operations with Polynomials:
  • Addition/Subtraction: Combine like terms.
  • Multiplication: Use the distributive property.
  • Division: Polynomial long division or synthetic division.
• Factoring Polynomials:
  • Common Factors: Factor out the 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, a^2 - 2ab + b^2 = (a - b)^2.
  • Factoring Quadratics: Express a quadratic trinomial as the product of two binomials.

Exponents and Radicals

• Exponents indicate repeated multiplication of a base.
• Laws 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^(mn).
  • Power of a Product: (ab)^n = a^n * b^n.
  • Power of a Quotient: (a/b)^n = a^n / b^n.
• Negative Exponents: a^(-n) = 1 / a^n.
• Zero Exponent: a^0 = 1 (if a ≠ 0).
• Radicals (roots) are the inverse operation of exponentiation.
• Square Root: √a is the number that, when multiplied by itself, equals a.
• nth Root: ⁿ√a is the number that, when raised to the nth power, equals a.
• Rational Exponents: a^(m/n) = ⁿ√a^m = (ⁿ√a)^m.
• Simplifying Radicals: Remove perfect square factors from under the radical sign.
• Operations with Radicals:
  • Addition/Subtraction: Combine like radicals (same radicand).
  • Multiplication: Multiply the radicands.
  • Division: Divide the radicands.
• Rationalizing the Denominator: Eliminate radicals from the denominator of a fraction.

Logarithms

• Logarithms are the inverse of exponential functions.
• Definition: log_b(x) = y if and only if b^y = x, where b is the base.
• Common Logarithm: Base 10 logarithm (log_10(x) or log(x)).
• Natural Logarithm: Base e logarithm (log_e(x) or ln(x)), where e ≈ 2.71828.
• Properties of Logarithms:
  • Product Rule: log_b(mn) = log_b(m) + log_b(n).
  • Quotient Rule: log_b(m/n) = log_b(m) - log_b(n).
  • Power Rule: log_b(m^p) = p * log_b(m).
  • Change of Base Formula: log_a(x) = log_b(x) / log_b(a).
• Solving Logarithmic Equations: Use the properties of logarithms to isolate the variable.
• Solving Exponential Equations: Take the logarithm of both sides to isolate the variable.