Algebra Overview 📊

Like Terms

• Like terms: have the same variable part
• Ex: 5x + 4x → 5x, 4x are like terms → 5x + 4x = 9x
• Combine coefficients of like terms:
  • 3x + 4y + 5x + 8y → 8x + 12y

Combining Radicals

• Radicals: Like terms must have the same radicand:
  • 3√2 + 8√2 + 5√7 + 3√7 → 11√2 + 8√7

Polynomial Operations

• Use properties to combine or distribute terms
• Addition/Subtraction:
  • Add/subtract coefficients of like terms
  • Ex: 7x + 4x^2 + 5x + 9x^2 → 13x^2 + 12x
  • Distribute negative signs
• Multiplication: Distribute each term
  • Ex: monomial x trinomial: 7x * (x² + 2x - 3) → 7x³ + 14x² - 21x
  • Ex: binomial x binomial: 3x - 4 * 2x + 7 → Apply FOIL method: 6x² + 13x - 28
  • Ex: binomial x trinomial: Distribute each term in the binomial across the trinomial

Properties of Exponents

• Multiplication: Add exponents
  • x³ * x⁴ = x⁷
• Division: Subtract exponents
  • x⁹ / x⁴ = x⁵
• Power of a Power: Multiply exponents
  • (x²)³ = x⁶
• Negative Exponents: Move base to the opposite part of the fraction
  • x^(-3) = 1/x³*

Simplifying Expressions:

• Apply multiplication, power rules, and combine like terms
  • Ex: (3x³y⁻²)(7x⁻⁸y⁵) → (21x⁻⁵y³) which becomes 21y³/x⁵

Solving Equations

• Linear equations:
  • Isolate x: Ex: 2x + 4 = 9 → x = 2.5
  • Combine like terms, move constants to one side, divide to solve for x
• Quadratic equations (factoring, quadratic formula):
  • Factoring: ax² + bx + c = 0
  • Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a)
  • Complete the square method for non-factorable cases

Graphing Linear Equations

• y = mx + b form: m = slope, b = y-intercept
• Plot y-intercept, use slope to find another point
• Standard Form: Ax + By = C
  • Convert to slope-intercept for graphing: y = mx + b

Writing Equations of Lines

• Given a point & slope: Use point-slope form
  • y - y1 = m(x - x1)
  • Convert to slope-intercept: y = mx + b
• Given 2 points: Find slope, use point-slope form
  • m = (y2 - y1) / (x2 - x1)

Parallel and Perpendicular Lines

• Parallel lines: Same slope
• Perpendicular lines: Negative reciprocal slopes (if m1 = a/b, m2 = -b/a)

Practice Problems

• Many practice problems given to apply and test understanding of each section.