Introduction to Algebra

• Algebra involves symbol manipulation based on defined rules.
• Symbols in algebra are variables, which represent quantities without fixed values.
• Algebra provides a foundation for almost all areas of mathematics.
• It ranges from solving equations to studying groups, rings, and fields.

Variables

• Variables are symbols, often letters, representing unknown or changeable values.
• They express relationships between quantities.
• In y = x + 2, x and y are variables.

Constants

• Constants are values that remain unchanged.
• They are typically numbers.
• In y = x + 2, 2 is a constant.

Expressions

• Expressions combine variables, constants, and mathematical operations.
• Expressions lack an equals sign.
• Examples include x + 2, 3y - 5, and a².

Equations

• Equations state the equality of two expressions.
• They include an equals sign.
• Examples include y = x + 2, 3x - 1 = 8, and a² + b² = c².

Coefficients

• A coefficient is a number multiplying a variable.
• In 3x, 3 is the coefficient of x.

Terms

• A term is a single number or variable, or a product of numbers and variables.
• Terms are separated by + or - signs within an expression or equation.
• In 2x + 3y - 5, 2x, 3y, and -5 are terms.

Operations

• Basic algebraic operations: addition, subtraction, multiplication, and division.
• Other operations: exponents and roots.

Order of operations

• The order of operations dictates the sequence of performing mathematical operations.
• PEMDAS/BODMAS is a common mnemonic.
  • Parentheses/Brackets come first
  • Exponents/Orders are next
  • Multiplication and Division follow (from left to right)
  • Addition and Subtraction are last (from left to right)

Simplifying expressions

• Simplifying rewrites an expression in its most basic form.
• This combines like terms and performs operations.
• Like terms share the same variable raised to the same power.
• To simplify 2x + 3x - y + 4y:
  • Combine like terms: (2x + 3x) + (-y + 4y) = 5x + 3y

Solving equations

• Solving finds the variable values that make the equation true.
• This involves isolating the variable.
• Inverse operations are used for isolation.
  • To solve x + 5 = 12
    • Subtract 5 from both sides: x + 5 - 5 = 12 - 5
    • Simplify: x = 7

Linear equations

• Linear equations have a maximum variable power of 1.
• They can be written as ax + b = c, where a, b, and c are constants.
• Solutions involve isolating x through inverse operations.

Quadratic equations

• Quadratic equations have a maximum variable power of 2.
• They can be written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Solution methods:
  • Factoring
  • Completing the square
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Systems of equations

• Systems of equations are sets of two or more equations with the same variables.
• Solutions are variable values satisfying all equations simultaneously.
• Solution methods:
  • Substitution
  • Elimination (addition/subtraction)
  • Graphing

Inequalities

• Inequalities compare two expressions using inequality symbols.
• Common symbols:
  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
• Solving is similar to equations, but:
  • Multiplying/dividing by a negative number reverses the inequality symbol.

Polynomials

• Polynomials include variables and coefficients with addition, subtraction, multiplication, and non-negative integer exponents.
• They can have one or more terms.
• Examples: x² + 3x + 2, 5y⁴ - 2y² + y - 7

Factoring polynomials

• Factoring expresses a polynomial as a product of simpler polynomials.
• Techniques:
  • Factoring out the greatest common factor (GCF)
  • 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
  • Factoring by grouping

Exponents

• An exponent indicates how many times a base is multiplied by itself.
• In x³, x is the base, and 3 is the exponent: x³ = x * x * x
• 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 (Any non-zero number to the power of 0 equals 1)
  • x⁻ᵃ = 1/xᵃ (Negative exponent)

Radicals

• Radicals involve roots like square roots, cube roots, etc.
• The radical symbol is √.
• Example: √9 = 3
• Simplifying involves removing perfect square factors:
  • √12 = √(4 * 3) = √4 * √3 = 2√3

Rational expressions

• Rational expressions are fractions with polynomials in the numerator and denominator.
• Operations:
  • Addition/subtraction requires a common denominator.
  • Multiplication multiplies numerators and denominators separately.
  • Dividing multiplies by the reciprocal of the divisor.
  • Simplifying factors the numerator and denominator and cancels common factors.

Functions

• Functions relate inputs to outputs, with each input having exactly one output.
• The input is the argument or independent variable (x); the output is the value or dependent variable (y).
• Function notation: y = f(x).
• Example: f(x) = 2x + 3

Graphing

• Graphing visually represents algebraic relationships on a coordinate plane.
• The coordinate plane has x (horizontal) and y (vertical) axes.
• Linear equations graph as straight lines.
• Quadratic equations graph as parabolas.

Slope

• Slope (m) measures a line's steepness and direction.
• It's the change in y divided by the change in x (rise over run).
• Formula: m = (y₂ - y₁) / (x₂ - x₁)
• Slope-intercept form: y = mx + b, where m is the slope, and b is the y-intercept.

Intercepts

• The x-intercept is where the graph crosses the x-axis (y = 0).
• The y-intercept is where the graph crosses the y-axis (x = 0).

Distance Formula

• Distance between (x₁, y₁) and (x₂, y₂): √((x₂ - x₁)² + (y₂ - y₁)²)

Midpoint Formula

• Midpoint between (x₁, y₁) and (x₂, y₂): ((x₁ + x₂) / 2, (y₁ + y₂) / 2)