Notes on Adding, Subtracting, and Multiplying Polynomial Expressions

Adding Polynomial Expressions

• Example 1: Add the polynomials
  • Expression: 4x² + 5x + 7 + 3x² - 8x + 12
  • Combine like terms:
    • 4x² + 3x² = 7x²
    • 5x - 8x = -3x
    • 7 + 12 = 19
  • Result: 7x² - 3x + 19

Subtracting Polynomial Expressions

• Example 2: Subtract the polynomials

  • Expression: 9x² - 7x + 13 - (5x² + 7x + 14)
  • Distribute the negative sign:
    • 9x² - 7x + 13 - 5x² - 7x - 14
  • Combine like terms:
    • 9x² - 5x² = 4x²
    • -7x + 7x = 0 (cancels out)
    • 13 - 14 = -1
  • Result: 4x² - 1

• Example 3: Another subtraction problem

  • Expression: 3x³ - 5x + 8 - (7x² + 6x - 9)
  • Distribute the negative sign:
    • 3x³ - 5x + 8 - 7x² - 6x + 9
  • Combine like terms:
    • 3x³ (only one)
    • -7x² (only one)
    • -5x - 6x = -11x
    • 8 + 9 = 17
  • Result: 3x³ - 7x² - 11x + 17

Multiplying Polynomial Expressions

Multiplying Binomials

• Example: Multiply 3x + 5 by 2x - 3
  • Use the FOIL method:
    • 3x * 2x = 6x²
    • 3x * -3 = -9x
    • 5 * 2x = 10x
    • 5 * -3 = -15
  • Combine like terms:
    • -9x + 10x = 1x
  • Result: 6x² + x - 15

Squaring a Binomial

• Example: Simplifying (2x - 5)²
  • FOIL method:
    • 2x * 2x = 4x²
    • 2x * -5 = -10x
    • -5 * 2x = -10x
    • -5 * -5 = 25
  • Combine like terms:
    • -10x - 10x = -20x
  • Result: 4x² - 20x + 25

Multiplying a Binomial by a Trinomial

• Example: Multiply 4x - 2 by x² + 3x - 5
  • Calculate:
    • 4x * x² = 4x³
    • 4x * 3x = 12x²
    • 4x * -5 = -20x
    • -2 * x² = -2x²
    • -2 * 3x = -6x
    • -2 * -5 = 10
  • Combine like terms:
    • Result: 4x³ + 10x² - 26x + 10

Multiplying Trinomials

• When multiplying a trinomial by another trinomial, expect to combine 9 terms initially.

Dividing Polynomial Expressions

Methods of Division

1. Factoring
2. Long Division
3. Synthetic Division

Example of Division by Factoring

• Divide: x² + 7x + 12 by x + 3
  • Factor: (x + 3)(x + 4)
  • Cancel: Result: x + 4

Example of Long Division

• Divide: 2x² - x + 6 by x - 2
  • Set up the long division:
    • Result: 2x - 3

Example of Synthetic Division

• Divide: 2x² - 7x + 6 by x - 2
  • Result from synthetic division: 2x - 3

Conclusion

• Various methods for manipulating polynomial expressions are critical in algebra. Practice with examples to master addition, subtraction, multiplication, and division.