Lecture Notes: Polynomial Factorization and Quadratics

Perfect Square Trinomials

• Objective: Convert a polynomial into the form ((a + b)^2).
• Example: (x^2 + 8x + 16)
  • Recognized as a perfect square because:
    • (x^2 = a^2)
    • (8x = 2ab (where (b = 4))
    • (16 = b^2)
  • Factorization: ((x + 4)^2)
• Challenge: Polynomials not in perfect square form

General Factoring of Quadratics

• Example: (x^2 + 10x + 16)
  • Not a perfect square ((x + 4)^2).
• Diamond Method:
  • Multiply coefficient of (x^2) by constant term: (1 \times 16 = 16).
  • Middle term is 10.
  • Find numbers (n) and (m) such that:
    • (n \times m = 16)
    • (n + m = 10)
  • Solution: (n = 8), (m = 2)
• Rewrite Polynomial:
  • (x^2 + 10x + 16 = x^2 + 8x + 2x + 16)
• Factor by Grouping:
  • Group terms: ((x^2 + 8x) + (2x + 16))
  • Factor out GCF:
    • (x(x + 8) + 2(x + 8))
  • Final Factorization: ((x + 2)(x + 8))

Advanced Example

• Problem: (12x^2 - 26x - 10)
• Initial Step: Factor out common factor (2):
  • (2(6x^2 - 13x - 5))
• Diamond Method for Inner Polynomial:
  • Multiply leading and constant coefficients: (6 \times -5 = -30)
  • Middle term: (-13)
  • Find (n) and (m):
    • (n \times m = -30)
    • (n + m = -13)
  • Solution: (n = -15), (m = 2)
• Rewrite:
  • (6x^2 - 13x - 5 = 6x^2 - 15x + 2x - 5)
• Factor by Grouping:
  • Group terms: ((6x^2 - 15x) + (2x - 5))
  • Factor out GCF:
    • (3x(2x - 5) + 1(2x - 5))
  • Final Factorization: (2(2x - 5)(3x + 1))

Conclusion

• Practice general method for complex quadratics.
• Ensure mastery of diamond method and factor by grouping.
• Next session: More factoring techniques and examples.