Factoring Trinomials with Leading Coefficient Not Equal to One

Introduction

Discusses a method for factoring trinomials where the leading coefficient is not one.
Useful technique for those looking to improve their skills in factoring, particularly relevant in the U.S.

Initial Step
- Take the leading coefficient (12) and multiply it by the constant term (6).
- Expression changes from 12x^2 + 17x + 6 to x^2 + 17x + 72.
Factoring Process
- Set up the factor format: (X + _)(X + _).
- Find two numbers that multiply to 72 and add up to 17.
- List of factor pairs for 72:
  - 1 and 72
  - 2 and 36
  - 3 and 24
  - 4 and 18
  - 6 and 12
  - 8 and 9
- Numbers that add up to 17: 8 and 9.
- Insert into the factor format: (X + 8)(X + 9).
Reverse Multiplication Adjustment
- Divide the factors 8 and 9 by the original leading coefficient (12):
  - Reduce fractions: (X + 2/3)(X + 3/4).
- Adjust by multiplying the denominator with X:
  - (3X + 2)(4X + 3).
Verification
- Expanding gives 12x^2 + 17x + 6, matching the original expression.

Initial Step
- Multiply leading coefficient of 6 by constant term, resulting in x^2 - 5X - 24.
Factoring Process
- Set up the factor format: (X + _)(X - _) because of the negative third term.
- Find two numbers that multiply to 24 and subtract to get 5.
- List of factor pairs for 24:
  - 1 and 24
  - 2 and 12
  - 3 and 8
  - 4 and 6
- Numbers that subtract to 5: 8 and 3.
- Insert into the factor format: (X - 8)(X + 3).
- Place larger number (8) with the negative sign due to -5 coefficient.
Reverse Multiplication Adjustment
- Divide the factors 3 and 8 by the original leading coefficient (6):
  - Reduce fractions: (X + 1/2)(X - 4/3).
- Adjust by multiplying the denominator with X:
  - (2X + 1)(3X - 4).
Verification
- Expanding gives 6x^2 - 5x - 4, confirming the factorization.

The method provides a systematic approach to factoring non-standard trinomials.
Encouragement to practice and apply this method for efficiency in solving such problems.