Introduction to Algebra

• Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols
• These symbols represent quantities without fixed values, known as variables
• Algebra is fundamental to nearly all other areas of mathematics, science, and engineering

Basic Operations

• Algebraic expressions use basic arithmetic operations: addition, subtraction, multiplication, and division
• These operations are performed on both constants (numbers) and variables (symbols representing numbers)
• Exponents indicate repeated multiplication

Variables and Constants

• Variables are symbols, usually letters, that represent unknown or changing quantities
• Constants are fixed numerical values
• In the expression "3x + 5", 'x' is the variable, and 3 and 5 are constants

Algebraic Expressions

• Algebraic expressions combine variables, constants, and operations
• Examples include 5x, 2y + 3, and a^2 - 4ab + b^2
• Terms are parts of an expression separated by addition or subtraction

Equations

• An equation is a statement that asserts the equality of two expressions
• It contains an equals sign (=)
• The goal is often to find the value(s) of the variable(s) that make the equation true (solving the equation)
• Example: x + 5 = 10

Solving Equations

• Solving equations involves isolating the variable on one side of the equation
• This is done by performing the same operations on both sides of the equation to maintain equality
• Basic techniques include adding, subtracting, multiplying, or dividing both sides by the same value.
• Inverse operations are used to isolate variables

Linear Equations

• A linear equation is an equation where the highest power of the variable is 1
• It can be written in the form ax + b = 0, where a and b are constants and x is the variable
• The graph of a linear equation is a straight line

Solving Linear Equations

• To solve a linear equation, simplify both sides by combining like terms
• Then, isolate the variable by using inverse operations
• Example: To solve 2x + 3 = 7, subtract 3 from both sides (2x = 4), then divide by 2 (x = 2)

Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations with the same variables
• The solution to a system is the set of values for the variables that satisfy all equations simultaneously
• Systems can have one solution, no solution, or infinitely many solutions

Methods for Solving Systems

• Substitution: Solve one equation for one variable, then substitute that expression into the other equation
• Elimination: Add or subtract multiples of the equations to eliminate one variable
• Graphing: Graph each equation and find the point(s) of intersection, which represent the solution(s)

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols (<, >, ≤, ≥)
• Solving inequalities involves similar techniques to solving equations, but with some key differences
• Multiplying or dividing by a negative number reverses the inequality sign

Graphing Inequalities

• Inequalities can be graphed on a number line or in a coordinate plane
• For one-variable inequalities, use a number line with open or closed circles to indicate whether the endpoint is included
• For two-variable inequalities, graph the boundary line and shade the region that satisfies the inequality

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Examples: x^2 + 3x - 4, 5y^3 - 2y + 1
• Polynomials are classified by their degree (the highest power of the variable) and number of terms

Operations with Polynomials

• Polynomials can be added, subtracted, multiplied, and divided
• Addition and subtraction involve combining like terms
• Multiplication involves using the distributive property
• Division can be done using long division or synthetic division

Factoring Polynomials

• Factoring is the process of expressing a polynomial as a product of simpler polynomials
• Common techniques include factoring out a greatest common factor (GCF), using difference of squares, perfect square trinomials, and grouping

Quadratic Equations

• A quadratic equation is a polynomial equation of degree 2
• The general form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Quadratic equations can have two, one, or no real solutions

Solving Quadratic Equations

• Factoring: Factor the quadratic expression and set each factor equal to zero
• Quadratic Formula: Use the formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the solutions
• Completing the Square: Rewrite the equation in the form (x + p)^2 = q and then solve for x

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output
• Functions are often denoted as f(x), where x is the input and f(x) is the corresponding output

Types of Functions

• 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, whose graph is a parabola
• Exponential Functions: f(x) = a^x, where a is a constant
• Logarithmic Functions: f(x) = log_a(x), the inverse of exponential functions

Graphing Functions

• Functions can be graphed on a coordinate plane by plotting points (x, f(x))
• The graph provides a visual representation of the function's behavior
• Key features to identify include intercepts, slope, vertex (for quadratics), and asymptotes

Domain and Range

• The domain of a function is the set of all possible input values (x-values)
• The range of a function is the set of all possible output values (f(x)-values)
• Domain and range can be restricted by the function's equation or context

Exponents and Radicals

• Exponents indicate repeated multiplication
• Radicals (roots) are the inverse operation of exponentiation
• Understanding exponent rules is crucial for simplifying expressions

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: (ab)^n = a^n * b^n
• Power of a Quotient: (a/b)^n = a^n / b^n
• Negative Exponent: a^(-n) = 1 / a^n
• Zero Exponent: a^0 = 1 (if a ≠ 0)

Simplifying Radicals

• Radicals can be simplified by factoring out perfect squares (or cubes, etc.) from the radicand
• Example: √20 = √(4 * 5) = √4 * √5 = 2√5

Rationalizing Denominators

• Rationalizing the denominator involves eliminating radicals from the denominator of a fraction
• Multiply the numerator and denominator by a suitable expression that will eliminate the radical in the denominator

Logarithms

• A logarithm is the inverse operation to exponentiation
• log_b(x) = y means b^y = x, where b is the base, x is the argument, and y is the exponent

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 Exponential and Logarithmic Equations

• Exponential equations can often be solved by taking the logarithm of both sides
• Logarithmic equations can be solved by exponentiating both sides

Complex Numbers

• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (i^2 = -1)
• 'a' is the real part, and 'b' is the imaginary part

Operations with Complex Numbers

• Addition/Subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• Division: Multiply the numerator and denominator by the conjugate of the denominator