Introduction to Algebra

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

Basic Concepts

• Variables are symbols (usually letters) representing unknown or changing quantities
• Constants are fixed values that do not change
• Expressions are combinations of variables, constants, and mathematical operations
• Equations are statements showing the equality between two expressions

Operations

• Addition: Combining terms, the sum of two or more quantities
• Subtraction: Finding the difference between two quantities
• Multiplication: Repeated addition, product of two or more quantities
• Division: Splitting a quantity into equal parts

Order of Operations

• Parentheses/Brackets first
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)
• Acronym: PEMDAS or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Terms

• Like terms: Terms with the same variables raised to the same powers can be combined
• Coefficients numerical factors of a term

Simplifying Expressions

• Combining like terms
• Applying the distributive property: a(b + c) = ab + ac
• Using the order of operations to reduce complexity

Solving Equations

• Isolating the variable on one side of the equation
• Performing the same operation on both sides to maintain equality
• Using inverse operations to cancel out terms

Linear Equations

• Have the form ax + b = c, where a, b, and c are constants and x is the variable
• Solving involves isolating x
• Example: 2x + 3 = 7 Solution: x = 2

Quadratic Equations

• Have the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Factoring, completing the square, or using the quadratic formula can be used to solve
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)

Systems of Equations

• 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, and graphing

Inequalities

• Mathematical statements comparing two expressions using symbols
• Solving inequalities involves similar techniques to solving equations, but with some differences
• Multiplying or dividing by a negative number reverses the inequality sign

Functions

• Relationship 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

Domain and Range

• Domain: Set of all possible input values (x)
• Range: Set of all possible output values (f(x))

Linear Functions

• Functions with a constant rate of change
• Represented as f(x) = mx + b, where m is the slope and b is the y-intercept

Polynomials

• Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents
• Can be classified by the highest degree of the variable
• Examples: Linear (degree 1), Quadratic (degree 2), Cubic (degree 3)

Factoring Polynomials

• Breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial
• Common techniques include factoring out a common factor, difference of squares, and trinomial factoring

Rational Expressions

• Fractions where the numerator and denominator are polynomials
• Simplifying involves factoring and canceling common factors
• Operations (addition, subtraction, multiplication, division) follow similar rules to numerical fractions

Exponents

• Indicate repeated multiplication of a base
• a^n means a multiplied by itself n times

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)

Radicals

• Represent the inverse operation of exponentiation
• The nth root of a is written as ⁿ√a

Simplifying Radicals

• Factoring out perfect squares (or cubes, etc.) from the radicand
• Rationalizing the denominator to remove radicals from the denominator of a fraction

Logarithms

• The inverse function to exponentiation
• log_b(a) = x means b^x = a

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_b(a) = log_c(a) / log_c(b)

Absolute Value

• The distance of a number from zero on the number line
• Denoted as |x|
• Always non-negative

Complex Numbers

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

Operations with Complex Numbers

• Addition/Subtraction: Combine real and imaginary parts separately
• Multiplication: Use the distributive property and the fact that i^2 = -1
• Division: Multiply the numerator and denominator by the conjugate of the denominator