Common Concepts in Algebra

Like Terms

• Like terms: terms with the same variable(s) raised to the same power.
  • Example: 5x and 4x are like terms.
  • To combine: Add coefficients (5x + 4x = 9x).
• Example with multiple variables: 3x + 4y + 5x + 8y
  • Combine like terms:
    • 3x + 5x = 8x
    • 4y + 8y = 12y
• Example with radicals: 3√2 + 5√7 + 8√2 + 3√7
  • Combine like terms:
    • 3√2 + 8√2 = 11√2
    • 5√7 + 3√7 = 8√7

Adding/Combining Polynomials

• Combine terms by grouping like terms.
  • Example: 9x² + 6x + 5 + 3x² - 5x - 9
    • 9x² + 3x² = 12x²
    • 6x - 5x = 1x
    • 5 - 9 = -4
    • Result: 12x² + x - 4
  • Example with distribution: 3x² + 7x - 4 - 8x² - 5x + 7
    • Distribute negative sign: 3x² + 7x - 4 - 8x² + 5x - 7
    • Combine like terms: -5x² + 12x - 11

Definitions

• Polynomial: Function with many terms.
• Monomial: One term (e.g., 8x or 5x²).
• Binomial: Two terms (e.g., 5x + 6).
• Trinomial: Three terms (e.g., x² + 6x + 5).

Multiplying Polynomials

• Distribute terms across each other.
  • Example: 7x(x² + 2x - 3)
    • 7x * x² = 7x³
    • 7x * 2x = 14x²
    • 7x * -3 = -21x
    • Result: 7x³ + 14x² - 21x
• Multiplying binomial by trinomial: 5x²(3x⁴ - 6x³ + 5x - 8)
  • Distribute terms: 15x⁶ - 30x⁵ + 25x³ - 40x²
• FOIL for binomials: (a + b)(c + d) = ac + ad + bc + bd
  • Example: (3x - 4)(2x + 7)
    • 3x * 2x = 6x²
    • 3x * 7 = 21x
    • -4 * 2x = -8x
    • -4 * 7 = -28
    • Combine: 6x² + 13x - 28

Exponent Rules

• Multiplication: x³ * x⁴ = x⁷ (add exponents).
• Division: x⁹ / x⁴ = x⁵ (subtract exponents).
• Power of a power: (x²)³ = x⁶ (multiply exponents).
• Examples: 24x^7y^(-2)/6x^4y^5 simplifies to 4x³y^(-7); move y to get 4x³/y⁷.

Solving Equations

• Single variable: Isolate x.
  • Example: x + 4 = 9 -> x = 5
  • Check: Plug x back in.
• Two-step examples:
  • 3x + 5 = 11: First subtract, then divide -> x = 2
  • Distribute then solve: 2(x - 1) + 6 = 10
• Complex examples involving distribution and combining like terms.
  • Example: 2/3x + 5 = 8: Isolate variable, clear fractions -> x = 4.5

Factoring and Quadratics

• Factor: ax² + bx + c
  • Example: x² - 5x + 6: Factor into (x - 2)(x - 3)
• If leading coefficient is not 1, adjust and find common factors.
  • Example: 2x² + 3x - 2: Use factoring by grouping.
• Quadratic formula for solving: x = (-b ± √(b² - 4ac)) / 2a

Graphing Linear Equations

• Slope-intercept form y = mx + b
  • m: Slope
  • b: Y-intercept
• Steps:
  • Plot y-intercept.
  • Use slope to find the next point.
  • Draw the line through the points.
• Standard form: Ax + By = C
  • Find x and y intercepts.

Writing Linear Equations

• Given a point and slope: Use point-slope form and convert to slope-intercept or standard form.
  • Example: Point (1, 3), slope 2 -> y - 3 = 2(x - 1) -> y = 2x + 1
  • Convert to standard: -2x + y = 1
• Parallel/Perpendicular lines: Use point and negative reciprocal/modification of slope.

Complex Fractions and Simplifying

• Simplify by eliminating negative exponents, convert expressions, and perform arithmetic operations.
• Example 3x/5 ÷ 7xy/9: Keep, change, flip strategy to simplify complex fractions.