Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• It is a generalization of arithmetic, where specific numerical values are used
• Algebra provides tools and techniques for solving mathematical problems and making generalizations in math and other sciences

Variables and Constants

• A variable is a symbol (usually a letter) that represents an unknown or changeable value
• Variables are used to express relationships between quantities that can vary
• A constant is a fixed value that does not change in a given context
• Constants are typically numbers, but can also be symbols representing fixed values

Algebraic Expressions

• An algebraic expression is a combination of variables, constants, and algebraic operations
• Examples of algebraic operations include addition, subtraction, multiplication, division, and exponentiation
• Algebraic expressions do not contain equality or inequality signs

Equations

• An equation is a statement that two algebraic expressions are equal
• Equations contain an equals sign "=" to show the equivalence of the expressions on either side
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true

Basic Operations

• Addition: Combining terms
• Subtraction: Finding the difference between terms
• Multiplication: Scaling one term by another
• Division: Splitting one term into equal parts based on another term
• Exponentiation: Raising a term to a power

Order of Operations

• The order of operations is a convention used to evaluate mathematical expressions consistently
• PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction (done from left to right)

Terms

• A term is a single number, variable, or a product of numbers and variables
• Like terms have the same variables raised to the same powers and can be combined through addition or subtraction
• Unlike terms have different variables or different powers and cannot be combined directly

Coefficients

• A coefficient is a numerical factor that multiplies a variable in a term
• For example, in the term 3x, 3 is the coefficient of x
• If a term has no explicit numerical factor, the coefficient is assumed to be 1

Polynomials

• A polynomial is an algebraic expression consisting of one or more terms, where each term contains only non-negative integer exponents
• Examples: x^2 + 2x + 1, 3y^4 - 7y^2 + 2y - 5
• Polynomials can be classified by the number of terms they contain:
  • Monomial: One term (e.g., 5x^2)
  • Binomial: Two terms (e.g., 2x + 3)
  • Trinomial: Three terms (e.g., x^2 - 4x + 7)

Degree of a Polynomial

• The degree of a term in a polynomial is the sum of the exponents of the variables in that term
• The degree of a polynomial is the highest degree of any term in the polynomial
• For example, the degree of x^3 + 2x^2 - 5x + 1 is 3

Factoring Polynomials

• Factoring is the process of expressing a polynomial as a product of simpler polynomials or factors
• Common factoring techniques include:
  • 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 and a^2 - 2ab + b^2 = (a - b)^2
  • Quadratic Trinomials: Factoring expressions of the form ax^2 + bx + c

Solving Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1
• To solve a linear equation, isolate the variable on one side of the equation by performing the same operations on both sides
• Common steps include:
  • Simplifying both sides of the equation
  • Adding or subtracting terms to move variables to one side and constants to the other
  • Multiplying or dividing to isolate the variable

Solving Quadratic Equations

• A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a ≠ 0
• Quadratic equations can be solved by:
  • Factoring
  • Completing the Square
  • Using the Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)

Systems of Equations

• A system of equations is a set of two or more equations containing the same variables
• To solve a system of equations, find the values of the variables that satisfy all equations simultaneously
• Common methods for solving systems of equations include:
  • Substitution
  • Elimination (Addition/Subtraction)
  • Graphing

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols
• Inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to)
• Solving inequalities involves finding the range of values that satisfy the inequality
• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed

Functions

• A function is 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
• The input is called the independent variable, and the output is called the dependent variable
• Functions are often represented using the notation f(x), where x is the input and f(x) is the output
• Common types of functions include:
  • Linear Functions: f(x) = mx + b
  • Quadratic Functions: f(x) = ax^2 + bx + c
  • Exponential Functions: f(x) = a^x
  • Logarithmic Functions: f(x) = log_a(x)

Graphing

• Graphing involves plotting points on a coordinate plane to visualize relationships between variables
• The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical)
• Points are represented by ordered pairs (x, y), where x is the horizontal coordinate and y is the vertical coordinate
• Graphs can be used to represent equations, functions, and data sets

Exponents and Radicals

• Exponents represent repeated multiplication of a base number
• Radicals represent the inverse operation of exponentiation, finding the root of a number
• Key rules for exponents:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(m*n)
  • a^0 = 1
  • a^(-n) = 1/a^n
• Key rules for radicals:
  • √(a*b) = √a * √b
  • √(a/b) = √a / √b
  • √a^2 = a (if a ≥ 0)

Rational Expressions

• A rational expression is a fraction where the numerator and denominator are polynomials
• Simplifying rational expressions involves factoring the numerator and denominator and canceling out common factors
• Operations with rational expressions:
  • Multiplication: Multiply numerators and denominators
  • Division: Multiply by the reciprocal of the divisor
  • Addition/Subtraction: Find a common denominator and combine numerators

Word Problems

• Word problems involve translating real-world scenarios into algebraic equations or expressions
• Steps for solving word problems:
  • Read the problem carefully and identify the unknowns
  • Assign variables to the unknowns
  • Write equations based on the given information
  • Solve the equations
  • Check the solution to make sure it makes sense in the context of the problem