Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• These symbols represent quantities without fixed values, known as variables.

Variables and Constants

• A variable is a symbol (usually a letter) that represents a value that can change or is unknown.
• A constant is a fixed value that does not change.

Expressions

• An algebraic expression is a combination of variables, constants, and algebraic operations (addition, subtraction, multiplication, division, exponentiation, etc.).
• Example: 3x + 2y - 5 is an algebraic expression.

Equations

• An equation is a statement that two expressions are equal.
• It contains an equals sign (=).
• Example: 3x + 2 = 5 is an equation.

Terms

• A term is a single number or variable, or numbers and variables multiplied together.
• Terms are separated by + or - signs in an expression.
• Like terms have the same variables raised to the same powers and can be combined.
• Example: In the expression 4x + 3y - 2x + 5, 4x and -2x are like terms.

Coefficients

• A coefficient is the numerical factor of a term that contains a variable.
• Example: In the term 4x, 4 is the coefficient.

Basic Operations

• Addition: Combining like terms.
• Subtraction: Removing or reducing quantities.
• Multiplication: Repeated addition or scaling.
• Division: Splitting into equal parts.

Order of Operations

• The order of operations (PEMDAS/BODMAS) dictates the sequence in which operations should be performed.
  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Simplifying Expressions

• Combining like terms to reduce the complexity of an expression.
• Using the distributive property to remove parentheses.
• Factoring: Expressing an expression as a product of its factors. Expanding: Multiplying out terms to remove parentheses.

Solving Equations

• The goal is to isolate the variable on one side of the equation to find its value.
• Use inverse operations to undo operations performed on the variable.
• Maintain balance by performing the same operation on both sides of the equation.

Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1.
• Standard form: ax + b = c, where a, b, and c are constants and x is the variable.
• Solving linear equations involves isolating the variable.

Quadratic Equations

• A quadratic equation is an equation in which the highest power of the variable is 2.
• Standard form: ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Methods for solving quadratic equations:
  • Factoring
  • Completing the square
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Systems of Equations

• A system of equations is a set of two or more equations with the same variables.
• Goal: find the values of the variables that satisfy all equations simultaneously.
• Methods for solving systems of equations:
  • Substitution
  • Elimination (Addition/Subtraction)
  • Graphing

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to).
• Solving inequalities is similar to solving equations, but multiplying or dividing by a negative number reverses the inequality sign.

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.
• Notation: f(x), where x is the input and f(x) is the output.

Domain and Range

• The domain of a function is the set of all possible input values (x).
• The range of a function is the set of all possible output values (f(x)).

Graphing Functions

• Functions can be represented graphically on a coordinate plane.
• The graph of a function shows the relationship between input values (x-axis) and output values (y-axis).

Exponents

• An exponent indicates how many times a base number is multiplied by itself.
• Rules of exponents:
  • xᵃ * xᵇ = xᵃ⁺ᵇ (Product of powers)
  • xᵃ / xᵇ = xᵃ⁻ᵇ (Quotient of powers)
  • (xᵃ)ᵇ = xᵃᵇ (Power of a power)
  • (xy)ᵃ = xᵃyᵃ (Power of a product)
  • (x/y)ᵃ = xᵃ/yᵃ (Power of a quotient)
  • x⁰ = 1 (Zero exponent)
  • x⁻ᵃ = 1/xᵃ (Negative exponent)

Radicals

• A radical is the root of a number (e.g., square root, cube root).
• The square root of x is written as √x.
• Simplifying radicals involves finding perfect square factors.

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: x² + 3x + 2, 5x⁴ - 2x² + x - 7.
• Polynomials can be classified by the number of terms:
  • Monomial: One term (e.g., 5x²)
  • Binomial: Two terms (e.g., x + 3)
  • Trinomial: Three terms (e.g., x² + 2x + 1)

Factoring Polynomials

• Factoring involves expressing a polynomial as a product of simpler polynomials.
• Common factoring techniques:
  • Greatest Common Factor (GCF)
  • Difference of Squares: a² - b² = (a + b)(a - b)
  • Perfect Square Trinomials: a² + 2ab + b² = (a + b)², a² - 2ab + b² = (a - b)²
  • Factoring by Grouping
  • Factoring Quadratics

Rational Expressions

• A rational expression is a fraction where the numerator and denominator are polynomials.
• Simplifying rational expressions involves factoring and canceling 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.