Lecture Notes: Factor by Grouping

Introduction

• Factor by grouping is a useful but not common technique.
• Commonly used for polynomials with four terms; can also be applied to five or six terms.

General Steps for Factor by Grouping

1. Check for a Greatest Common Factor (GCF)
   • Look for a common factor in all terms.
   • If no GCF, proceed to grouping.
2. Group Terms
   • Typically, group the first two and last two terms.
   • Keep the plus sign in the middle; a minus will require additional steps.
3. Factor Each Group
   • Find a common factor in each group separately.
   • Factor out the common elements.
4. Check for Common Binomial Factor
   • The expressions in the parentheses should be identical.
   • If identical, factor out the common binomial.

Example Process

1. Given terms: m^2 + 2m + mn + 2n
2. Group: (m^2 + 2m) + (mn + 2n)
3. Factor out GCFs:
   • First group: m(m + 2)
   • Second group: n(m + 2)
4. Common factor: m + 2
5. Resulting factorization: (m + 2)(m + n)
6. Verification by multiplication results in the original expression.

Handling Negative Signs

• If a minus is between groups, adjust by converting subtraction to adding a negative.
• Example: 7y - 9x - 3xy + 21
  • Rearrange terms as needed to make factor by grouping possible.
  • Ensure a plus between parentheses by changing - to + (-).
  • Re-attempt grouping and factoring.

Example with Rearrangement

• Terms: 7y + 21 - 9x - 3xy
• Group and factor:
  • First group: 7(y + 3)
  • Second group: -3x(y + 3)
• Common factor of (y + 3)
• Result: (y + 3)(7 - 3x)
• Verification by expanding leads back to the original terms.

Tips for Success

• Always rearrange terms if initial grouping doesn’t work.
• Ensure all operations correctly account for signs, especially when converting subtraction to addition.
• Verification through expansion helps confirm factorization accuracy.

Conclusion

• Factor by grouping can be tricky, especially with negative signs or when term order needs adjustment.
• Practice and familiarization with these steps enhance proficiency in using this technique.