Introduction to Algebra

• Algebra utilizes symbols to represent numbers and quantities
• Algebra provides a framework for generalizing arithmetic operations

Variables

• A variable, typically a letter, represents an unknown or changeable value
• Common variables include x, y, z, a, b, and c
• Variables help expressing relationships and solving for unknowns

Constants

• A constant is a fixed value that remains unchanged
• Constants can be numbers like 2, -5, π, and e

Expressions

• A combination of variables, constants, and arithmetic operations (+, -, ×, ÷) forms an algebraic expression
• Examples of expressions include: 3x + 2, y^2 - 4x + 7, and (a + b) / c

Equations

• An equation indicates the equality between two expressions
• Equations use an equals sign (=)
• Examples of equations: 2x + 5 = 11, x^2 - 3x + 2 = 0, and y = mx + b

Terms

• A single number, variable, or product of numbers and variables make up a term
• The terms in the expression 4x^2 - 2x + 7 are 4x^2, -2x, and 7

Coefficients

• A coefficient is the numerical factor of a term containing variables
• The coefficient in the term 4x^2 is 4
• The coefficient in the term -2x is -2

Like Terms

• Terms that possess the same variables raised to the same powers are like terms
• Examples of like terms: 3x and -5x; 2y^2 and 7y^2
• Unlike terms example: 4x and 4x^2 because the powers are different

Combining Like Terms

• Like terms are combined through addition or subtraction of their coefficients
• Example: 3x + 5x = 8x and 7y^2 - 2y^2 = 5y^2

Simplifying Expressions

• Combining like terms and performing operations simplifies an expression to its simplest form
• Example: 2(x + 3) - x = 2x + 6 - x = x + 6

Solving Equations

• Solving an equation involves determining the value(s) of the variable(s) that make the equation true
• This involves isolating the variable on one side of the equation

Addition and Subtraction Properties of Equality

• If a = b, then a + c = b + c (adding the same quantity to both sides)
• If a = b, then a - c = b - c (subtracting the same quantity from both sides)
• Adding or subtracting the same value from both sides maintains the equation's balance

Multiplication and Division Properties of Equality

• If a = b, then a × c = b × c (multiplying both sides by the same quantity)
• If a = b, then a / c = b / c (dividing both sides by the same non-zero quantity)
• Multiplying or dividing both sides by the same value (excluding division by zero) keeps the equation balanced

Linear Equations

• A linear equation features a variable with the highest power of 1
• Standard form: ax + b = c, where a, b, and c are constants, and x is the variable

Solving Linear Equations

• Isolate the variable using inverse operations
• As an example: 2x + 3 = 7 => 2x = 4 => x = 2

Systems of Linear Equations

• A system of linear equations includes two or more linear equations with the same variables
• The solution simultaneously satisfies all equations in the system

Methods for Solving Systems of Linear Equations

• Substitution: Solve for one variable in one equation, then substitute that expression into the other equation
• Elimination (Addition/Subtraction): Eliminate one variable by adding or subtracting multiples of the equations

Quadratic Equations

• In a quadratic equation, the highest power of the variable is 2
• Standard form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0

Methods for Solving Quadratic Equations

• Factoring: Express the quadratic expression as a product of two linear factors
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Completing the Square: Transform the equation into the form (x - h)^2 = k and solve for x

Factoring

• Factoring expresses a number or algebraic expression as a product of its factors
• Example: 12 = 2 × 2 × 3; x^2 - 4 = (x + 2)(x - 2)

Special Factoring Patterns

• Difference of Squares: a^2 - b^2 = (a + b)(a - b)
• Perfect Square Trinomial: a^2 + 2ab + b^2 = (a + b)^2
• Perfect Square Trinomial: a^2 - 2ab + b^2 = (a - b)^2

Polynomials

• Polynomials contain variables and coefficients, and use addition, subtraction, multiplication, and non-negative integer exponents
• Example: 3x^3 - 5x^2 + 2x - 7

Degree of a Polynomial

• The degree corresponds to the highest power of the variable in the polynomial
• The degree of 3x^3 - 5x^2 + 2x - 7 is 3

Operations with Polynomials

• Addition: Combine like terms of the polynomials
• Subtraction: Distribute the negative sign and combine like terms
• Multiplication: Use the distributive property to multiply each term of one polynomial by each term of the other polynomial

Exponents

• An exponent indicates how many times a base is multiplied by itself
• a^n = a × a × ... × a (n times), a is the base and n is the exponent

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
• Zero Exponent: a^0 = 1 (if a ≠ 0)
• Negative Exponent: a^(-n) = 1 / a^n

Radicals

• A radical represents a root of a number
• The square root, denoted by √, is the most common radical
• The nth root of a is denoted by ⁿ√a

Simplifying Radicals

• Identify perfect square factors (or perfect nth power factors for nth roots)
• Example: √20 = √(4 × 5) = √4 × √5 = 2√5

Rationalizing the Denominator

• Eliminate radicals from a fraction's denominator
• Multiply the numerator and denominator by a suitable expression, commonly the conjugate of the denominator

Functions

• A function relates inputs to permissible outputs, where each input corresponds to exactly one output
• Denoted as f(x), where x is the input and f(x) is the output

Domain and Range

• A function's domain consists of all possible input values (x-values)
• A function's range consists of all possible output values (f(x)-values)

Graphing Functions

• A graph provides a visual representation of the relationship between input and output values
• Plot points (x, f(x)) on a coordinate plane

Linear Functions

• A linear function's graph forms a straight line
• Standard form: f(x) = mx + b, where m is the slope and b is the y-intercept

Slope-Intercept Form

• y = mx + b, where m is the slope and b is the y-intercept
• Slope indicates the rate of change of the function
• The y-intercept is the point where the line crosses the y-axis

Point-Slope Form

• y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line
• Useful for determining the equation of a line given a point and the slope

Parallel and Perpendicular Lines

• Parallel lines share the same slope
• Perpendicular lines have slopes that are negative reciprocals (m1 × m2 = -1)

Inequalities

• An inequality uses inequality symbols (<, >, ≤, ≥) to compare two expressions
• Examples: x > 3, 2y + 1 ≤ 7

Solving Inequalities

• Employ the same techniques as solving equations, but reverse the inequality sign when multiplying or dividing by a negative number
• Example: -2x < 6 => x > -3

Absolute Value

• A number's absolute value represents its distance from zero
• Denoted as |x|
• |x| = x if x ≥ 0, and |x| = -x if x < 0

Solving Absolute Value Equations

• Isolate the absolute value expression
• Create two equations: one where the expression inside the absolute value equals the positive value, and the other where it equals the negative value
• Example: |x - 2| = 3 => x - 2 = 3 or x - 2 = -3, leading to x = 5 or x = -1