Algebra Final Exam Review

FOIL Method

• Objective: Multiply two binomials using the FOIL method.
• Example: (3x + 5)(4x - 7)
  • First: 3x * 4x = 12x²
  • Outside: 3x * (-7) = -21x
  • Inside: 5 * 4x = 20x
  • Last: 5 * (-7) = -35
  • Combine: 12x² - x - 35

Polynomial Difference

• Objective: Simplify the difference of two polynomials.
• Process:
  • Remove parentheses and distribute any negative signs.
  • Combine like terms.
• Example:
  • 9x³ - 5x² + 7 - (4x³ - 8x - 9)
  • Result: 5x³ - 5x² + 8x + 16

Slope Calculation

• Objective: Find the slope between two points.
• Formula: (y₂ - y₁)/(x₂ - x₁)
• Example: Points (3, -5) and (-9, -1)
  • Slope = 4 / -12 = -1/3

Order of Operations (PEMDAS)

• Steps:
  • Parentheses
  • Exponents
  • Multiplication/Division
  • Addition/Subtraction
• Example:
  • 8 + 3(2 - 9)²
  • Result: 155

Factoring Expressions

1. Difference of Squares: a² - b² = (a+b)(a-b)
   • Example: 9x² - 64
   • Result: (3x + 8)(3x - 8)
2. Trinomials: Find two numbers that multiply to the last term and add to the middle coefficient.
   • Example: x² + 7x + 12
   • Result: (x + 3)(x + 4)
3. Complex Trinomials: Leading coefficient different from 1.
   • Example: 6x² + x - 15
   • Result: (3x + 5)(2x - 3)
4. Perfect Cubes: a³ - b³ = (a - b)(a² + ab + b²)
   • Example: 27x³ - 64
   • Result: (3x - 4)(9x² + 12x + 16)

Solving Equations

• Isolate Variables: Move variables and constants across the equation.
• Example: 3x - 5 = 5x + 3
  • Result: x = 9

Graphing Linear Equations

1. Vertical Lines: x = a (line at x = a)
2. Horizontal Lines: y = b (line at y = b)
3. Standard Form: Find intercepts (x-intercept and y-intercept)
4. Slope-Intercept Form: y = mx + b
   • Start at y-intercept, use slope to find second point.

Quadratic Equations

• Methods: Factoring, Quadratic Formula
• Quadratic Formula: x = (-b ± √(b²-4ac)) / 2a

System of Equations

• Methods: Substitution, Elimination
• Example: Solve for x and y
  • Use elimination to cancel out variables.

Additional Topics

• Perimeter and Area of rectangles
  • Example: Length 10, Width 6, Area = 60, Perimeter = 32

Practice Resources

• YouTube Algebra Playlist: Search "new algebra playlist organic chemistry tutor" for more examples.