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.

Basic Concepts

• Variable: A symbol (usually a letter) representing an unknown or changeable value.
• Constant: A fixed value that does not change.
• Expression: A combination of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation). Example: 3x + 5.
• Equation: A statement that asserts the equality of two expressions, connected by an equals sign (=). Example: 3x + 5 = 14.
• Coefficient: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression. In the expression 3x, 3 is the coefficient.
• Term: Parts of an algebraic expression separated by + or - signs. In the expression 3x + 5, 3x and 5 are terms.

Operations

• Addition (+): Combining terms.
• Subtraction (-): Finding the difference between terms.
• Multiplication (* or ×): Repeated addition or scaling.
• Division (/ or ÷): Splitting a quantity into equal parts.
• Exponentiation (^): Raising a base to a power. x^2 means x squared.

Order of Operations

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

Solving Equations

• The goal is to isolate the variable on one side of the equation.
• Use inverse operations to undo operations performed on the variable.
• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.
• Whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality.
• Example: Solve for x in the equation 3x + 5 = 14.
• Subtract 5 from both sides: 3x = 9.
• Divide both sides by 3: x = 3.

Linear Equations

• An equation where the highest power of the variable is 1.
• General form: ax + b = c, where a, b, and c are constants and x is the variable.
• Solving involves isolating the variable x using inverse operations.

Quadratic Equations

• An equation where the highest power of the variable is 2.
• General form: ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Methods for solving:
  • Factoring: Expressing the quadratic expression as a product of two linear factors.
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a).
  • Completing the Square: Transforming the equation into the form (x + p)^2 = q.

Factoring

• The process of breaking down an expression into a product of its factors.
• Common techniques:
  • Greatest Common Factor (GCF): Finding the largest factor common to all terms and factoring it out.
  • 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.
  • Factoring by Grouping: Grouping terms to find common factors.

Systems of Equations

• A set of two or more equations with the same variables.
• Solution: A set of values for the variables that satisfies all equations in the system simultaneously.
• Methods for solving:
  • Substitution: Solving one equation for one variable and substituting that expression into the other equation.
  • Elimination (or Addition): Adding or subtracting the equations to eliminate one of the variables.
  • Graphing: Finding the point(s) of intersection of the graphs of the equations.

Inequalities

• Mathematical statements that compare 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: Similar to solving equations, but with some key differences.
• Multiplying or dividing by a negative number reverses the inequality sign.
• The solution to an inequality is often a range of values.

Polynomials

• An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• General form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_i are constants and n is a non-negative integer.
• Degree of a polynomial: The highest power of the variable.
• Types of polynomials:
  • Monomial: A polynomial with one term (e.g., 5x^2).
  • Binomial: A polynomial with two terms (e.g., 3x + 2).
  • Trinomial: A polynomial with three terms (e.g., x^2 + 2x + 1).
• Operations with polynomials:
  • Addition and Subtraction: Combining like terms.
  • Multiplication: Distributing each term of one polynomial to each term of the other polynomial.

Rational Expressions

• A fraction where the numerator and/or denominator are polynomials.
• Simplifying rational expressions: Factoring the numerator and denominator and canceling common factors.
• Operations with rational expressions:
  • Multiplication: Multiplying the numerators and the denominators.
  • Division: Multiplying by the reciprocal of the divisor.
  • Addition and Subtraction: Finding a common denominator and combining the numerators.

Exponents and Radicals

• Exponent: Indicates how many times a base is multiplied by itself. x^n means x multiplied by itself n times.
• Rules of exponents:
  • x^m * x^n = x^(m+n)
  • x^m / x^n = x^(m-n)
  • (x^m)^n = x^(m*n)
  • x^0 = 1 (if x ≠ 0)
  • x^(-n) = 1 / x^n
• Radical: A root of a number. √x represents the square root of x.
• Rules of radicals:
  • √(a*b) = √a * √b
  • √(a/b) = √a / √b
  • (√a)^2 = a

Functions

• 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) represents the output of the function f when the input is x.
• 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.
  • Exponential functions: f(x) = a^x.
  • Logarithmic functions: f(x) = log_b(x).
• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of all possible output values (y-values) that the function can produce.