Introduction to Algebra

• Algebra involves symbols and manipulation rules.
• Symbols usually represent variables, which are quantities without fixed values.
• Algebra, number theory, geometry, and analysis are major areas of mathematics.

Basic Algebraic Operations

• Algebra builds on addition, subtraction, multiplication, and division.
• Symbols in algebraic operations represent unknown values for equations and inequalities.
• Algebraic operations follow specific rules to solve for unknowns and simplify expressions.

Variables and Constants

• Variables are symbols (usually letters) representing unknown or changeable values.
• Constants are fixed values.
• In 3x + 5, x is a variable, and 3 and 5 are constants.

Expressions and Equations

• An algebraic expression combines variables, constants, and algebraic operations (e.g., 3x + 5).
• An algebraic equation shows the equality of two expressions (e.g., 3x + 5 = 14).

Solving Equations

• Solving an equation means finding variable values that make the equation true.
• Isolate the variable using inverse operations.
• To solve 3x + 5 = 14:
  • Subtract 5 from both sides: 3x = 9
  • Divide both sides by 3: x = 3
• x = 3 is the solution.

Linear Equations

• In a linear equation, the variable's highest power is 1.
• They are written as ax + b = 0, where a and b are constants and x is the variable.
• Basic algebraic operations isolate the variable to solve linear equations.

Quadratic Equations

• The highest power of the variable is 2 in a quadratic equation.
• The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Quadratic equations can be solved by factoring, using the quadratic formula, or completing the square.
• The quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
• Completing the square transforms the equation to (x + p)² = q.

Systems of Equations

• A system of equations is two or more equations with the same variables.
• The solution satisfies all equations simultaneously.
• Solving methods: substitution, elimination, and graphing.
• Substitution: solve for one variable and substitute.
• Elimination: add/subtract multiples of equations to eliminate a variable.
• Graphing: find intersection points of the equations' graphs.

Inequalities

• Inequalities compare two expressions using symbols like <, >, ≤, and ≥.
• Solving an inequality finds the variable value range that makes the inequality true.
• Reverse the inequality sign when multiplying/dividing by a negative number.
• Example: Solve -2x < 6:
  • Divide by -2 (and reverse the sign): x > -3

Polynomials

• Polynomials have variables and coefficients with addition, subtraction, multiplication, and non-negative integer exponents.
• Polynomials can be monomials (one term), binomials (two terms), trinomials (three terms), or have more terms.
• Monomial: 5x²
• Binomial: 2x + 3
• Trinomial: x² - 4x + 7

Factoring Polynomials

• Factoring polynomials expresses them as products of simpler polynomials.
• Common techniques: finding the greatest common factor (GCF), difference of squares, perfect square trinomials, and factoring by grouping.
• Difference of squares: a² - b² = (a + b)(a - b)
• Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²

Rational Expressions

• Rational expressions are fractions with polynomials in the numerator and denominator.
• Simplifying involves canceling common factors.
• Operations (addition, subtraction, multiplication, division) follow the same rules as numerical fractions.

Exponents and Radicals

• Exponents indicate the number of times a base is multiplied by itself.
• Rules of exponents:
  • xᵃ * xᵇ = xᵃ⁺ᵇ
  • (xᵃ)ᵇ = xᵃᵇ
  • xᵃ / xᵇ = xᵃ⁻ᵇ
  • x⁰ = 1
  • x⁻ᵃ = 1/xᵃ
• Radicals (roots) are the inverse of exponents.
• The nth root of x is a value that equals x when raised to the nth power.
• Example: √9 = 3 because 3² = 9.

Functions

• A function relates inputs to permissible outputs, with each input having exactly one output.
• The input is the function's argument, and the output is the function's value.
• Functions are denoted by symbols like f(x), where x is the input.

Graphing Functions

• Functions can be graphically represented on a coordinate plane.
• The graph of f(x) includes all points (x, f(x)).
• Common function types:
  • Linear functions: f(x) = mx + b (straight line)
  • Quadratic functions: f(x) = ax² + bx + c (parabola)
  • Exponential functions: f(x) = aˣ (exponential growth or decay)

Logarithms

• Logarithms are the inverse of exponentiation.
• The logarithm of x to base b is the exponent to which b must be raised to produce x: log_b(x) = y, meaning b^y = x.
• Common types:
  • Common logarithm: base 10 (log₁₀(x), often written as log(x))
  • Natural logarithm: base e (logₑ(x), written as ln(x))

Absolute Value

• The absolute value is a number's distance from zero.
• Denoted by vertical bars: |x|.
• |x| = x if x ≥ 0
• |x| = -x if x < 0
• Absolute value equations/inequalities require considering positive and negative cases.