Factoring Trinomials Lecture Notes

Introduction to Factoring Trinomials

• Objective: Understand how to factor trinomials where the leading coefficient is 1 and when it's not.
• Trinomial Example: Find two numbers that multiply to the constant term and add to the middle coefficient.

Factoring Trinomials with Leading Coefficient 1

Example 1: Factoring a Simple Trinomial

• Given: x^2 + 7x + 12
• Steps:
  • List factors of 12: (1, 12), (2, 6), (3, 4)
  • Find pair adding to 7: (3, 4)
  • Solution: (x + 3)(x + 4)

Example 2: x^2 + 11x + 30

• Steps:
  • Factors of 30: (1, 30), (2, 15), (3, 10), (5, 6)
  • Pair adding to 11: (5, 6)
  • Solution: (x + 5)(x + 6)

Solving Quadratic Equations

Example 3: x^2 - 5x + 6 = 0

• Steps:
  • Factors of 6: (1, 6), (2, 3)
  • Pair adding to -5: (-2, -3)
  • Factor: (x - 2)(x - 3)
  • Solve: Set each factor to zero
    • x - 2 = 0 → x = 2
    • x - 3 = 0 → x = 3

Example 4: x^2 + 3x - 28 = 0

• Steps:
  • Factors of -28: (-1, 28), (-2, 14), (-4, 7)
  • Pair adding to 3: (7, -4)
  • Factor: (x + 7)(x - 4)
  • Solve: x = -7, x = 4

Factoring Trinomials with Leading Coefficient Not Equal to 1

Example 5: 2x^2 - 7x + 6

• Steps:
  • Multiply leading term by constant: 2 * 6 = 12
  • Factors of 12: (3, 4)
  • Use negatives for correct sum: (-3, -4)
  • Grouping: 2x^2 - 4x - 3x + 6
  • Factor by Grouping: (x - 2)(2x - 3)
• Check by FOIL*

Example 6: 3x^2 - 7x - 6

• Steps:
  • Multiply leading term by constant: 3 * -6 = -18
  • Pair adding to -7: (2, -9)
  • Grouping: 3x^2 + 2x - 9x - 6
  • Factor by Grouping: (x - 3)(3x + 2)
• Check by FOIL*

Using the Quadratic Formula for Hard Trinomials

Example 7: 72x^2 + 17x - 70

• Steps:
  • Multiply leading term by constant: 72 * -70 = -5040
  • Difficult to factor manually
  • Use Quadratic Formula:
    • a = 72, b = 17, c = -70
    • Formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a
  • Solutions: x = 7/8, x = -10/9
  • Reverse Factor: (8x - 7)(9x + 10)
• Check by expanding*

Conclusion

• Skills Learned:
  • Factoring trinomials with different leading coefficients
  • Solving quadratic equations
  • Using quadratic formula for complex cases
• Practice: Apply learned methods to different quadratic equations.