Factoring Perfect Square Trinomials

Introduction

• Focus on factoring perfect square trinomials.

Example 1: x² + 6x + 9

• Identifying the numbers:
  • Two numbers that multiply to 9 and add to 6.
  • Numbers: 3 and 3.
• Factoring:
  • Factorization: (x + 3)(x + 3) = (x + 3)².
• Verification:
  • x² + 3x + 3x + 9 = x² + 6x + 9.
• Perfect square trinomial criteria:
  • The square roots of the first and last terms are 1 and 3.
  • Their product (3) is half of the middle term (6).

Example 2: x² + 10x + 25

• Identifying the numbers:
  • Square roots: 1 (x²) and 5 (25).
  • Product: 1 * 5 = 5, which is half of 10.
• Factoring:
  • Factorization: (x + 5)(x + 5) = (x + 5)².
• Verification using formula:
  • Formula: a² + 2ab + b² = (a + b)².
  • Here, a is x and b is 5.
  • Result: (x + 5)².*

Example 3: 4x² + 12x + 9

• Identifying the numbers:
  • Square roots: 2 (4) and 3 (9).
  • Product: 2 * 3 = 6, half of 12.
• Factoring:
  • Factorization: (2x + 3)².
• Different technique for leading coefficient not 1:
  • Multiply leading coefficient (4) and last coefficient (9): 4 * 9 = 36.
  • Find numbers that multiply to 36 and add to 12: 6 and 6.
  • Replace middle term: 4x² + 6x + 6x + 9.
  • Factor by grouping:
    • From first two: 2x(2x + 3).
    • From last two: 3(2x + 3).
  • Result: (2x + 3)².

Example 4: 9a² + 30ab + 25b²

• Identifying the numbers:
  • Square roots: 3 (9) and 5 (25).
  • Product: 3 * 5 = 15, which is half of 30.
• Factoring:
  • Factorization: (3a + 5b)².
• Verification using multiplication:
  • Multiply leading coefficient (9) and last coefficient (25): 9 * 25 = 225.
  • Find numbers that multiply to 225 and add to 30: 15 and 15.
  • Replace middle term: 9a² + 15ab + 15ab + 25b².
  • Factor by grouping:
    • First two: 3a(3a + 5b).
    • Last two: 5b(3a + 5b).
  • Result: (3a + 5b)².

Conclusion

• Factoring perfect square trinomials can be done through identifying patterns, using square roots, products, and factoring techniques.