Simplifying Algebraic Expressions

Combining Like Terms

• Example 1: Simplify 5x + 8 - 2x + 5

  • Combine like terms (5x and -2x; 8 and 5)
  • Calculation: 5x - 2x = 3x; 8 + 5 = 13
  • Simplified Expression: 3x + 13

• Example 2: Simplify 3x + 5y - 9x + 7y

  • Like terms: 3x and -9x; 5y and 7y
  • Calculation: 3 - 9 = -6x; 5 + 7 = 12y
  • Simplified Expression: -6x + 12y or 12y - 6x

• Example 3: Simplify 5xy + 6x² + 8xy - 9x²

  • Like terms: 5xy and 8xy; 6x² and -9x²
  • Calculation: 5 + 8 = 13xy; 6 - 9 = -3x²
  • Simplified Expression: 13xy - 3x²

The Distributive Property

• Example 1: Simplify 9(5x + 4)

  • Distribute: 9 * 5x = 45x; 9 * 4 = 36
  • Simplified Expression: 45x + 36

• Example 2: Simplify 7(3x² - 8x + 2)

  • Distribute: 7 * 3x² = 21x²; 7 * -8x = -56x; 7 * 2 = 14
  • Simplified Expression: 21x² - 56x + 14*

Combining Distributive Property and Like Terms

• Example: Simplify 5(3x + 4) - 7x + 8
  • Distribute: 5 * 3x = 15x; 5 * 4 = 20
  • Rewrite: 15x + 20 - 7x + 8
  • Combine Like Terms: 15x - 7x = 8x; 20 + 8 = 28
  • Simplified Expression: 8x + 28

Working with Polynomials

• Example: Simplify 3x² + 5x - 6 - (4x² - 7x - 3)
  • Remove Parentheses: Distribute -1 to the second polynomial
  • Combine Like Terms:
    • 3x² - 4x² = -x²
    • 5x - 7x = -2x
    • -6 + 3 = -3
  • Simplified Expression: -x² - 2x - 3

Multiplying Monomials

• x² * x³ = x⁵ (add exponents: 2 + 3 = 5)
• Examples: x⁴ * x⁷ = x¹¹; x³ * x⁻⁸ = 1/x⁵; x⁵ * x⁻³ = x²

Dividing Monomials

• x⁸ / x³ = x⁵ (subtract exponents: 8 - 3 = 5)
• Examples: x⁹ / x⁴ = x⁵; x³ / x¹¹ = 1/x⁸; x⁻⁴ / x⁵ = 1/x⁹

Simplifying Expressions with Multiple Terms

• Example: Simplify (36x⁷ / 4x³)
  • Divide Constants: 36 / 4 = 9
  • Divide Variables: x⁷ / x³ = x⁴
  • Simplified Expression: 9x⁴

FOIL Method (Multiplying Binomials)

• Example: (3x + 5)(2x - 3)
  • First: 3x * 2x = 6x²
  • Outer: 3x * -3 = -9x
  • Inner: 5 * 2x = 10x
  • Last: 5 * -3 = -15
  • Combine Like Terms: 6x² + x - 15

Squaring Binomials

• Example: (5x - 4)²
  • Expand: (5x - 4)(5x - 4)
  • Use FOIL:
    • 5x * 5x = 25x²
    • 5x * -4 = -20x
    • -4 * 5x = -20x
    • -4 * -4 = 16
  • Combine Like Terms: 25x² - 40x + 16

Multiplying Trinomials

• Example: (2x² - 7x + 4)(3x² + 5x + 7)
  • Apply Distribution and Combine Like Terms
  • Result: 6x⁴ - 11x³ - 9x² - 29x + 28

These notes cover the key points and methods for simplifying algebraic expressions, combining like terms, using the distributive property, and multiplying polynomials.