Expanding Binomials Lecture Notes

Goal

• Objective: Multiply binomials using the double distributive law and simplify more complicated polynomial expressions.

Terminology

• Double Distributive Law: A method to expand binomials, often referred to as FOIL (First, Outside, Inside, Last) in textbooks.
• Big Thumb Thing: An alternative teaching method using a thumb to cover terms for simplification.

Expanding Binomials

• When multiplying two sets of brackets, use the distributive law twice.
• Example: (2x + 3)(x - 2)
  • Cover one term with the thumb and multiply through:
    • 2x by x gives 6x²
    • 2x by -2 gives -4x
  • Move the thumb, multiply the next:
    • 3 by x gives +12x
    • 3 by -2 gives -8
• Simplification:
  • Combine like terms: 6x² + 8x - 8

FOIL Method

• FOIL stands for:
  • F: First terms
  • O: Outside terms
  • I: Inside terms
  • L: Last terms
• Use FOIL to multiply and simplify binomials but note its limitations with more complex expressions.

Complex Binomial Expansion

• For expressions beyond binomials like multiple terms:
  • Use double distributive law rather than FOIL.
  • Example: x(2x² + 4x - 1) + 3(2x² + 4x - 1)
    • Multiply x through the terms, then 3:
    • Combine like terms: 2x³ + 10x² + 11x - 3

Special Cases

• Squared Binomials: (2x + 3)²

  • Treat as (2x + 3)(2x + 3) and expand using double distributive law.
  • Example simplification: 4x² + 12x + 9

• Expression with Constants: 2(4x + 5)(x - 1)

  • Ignore constants initially, simplify inside brackets, then multiply through.
  • Example: 40x² + 6x + 4

Handling Subtraction in Brackets

• Manage sign changes carefully, especially with subtraction (e.g., minus a set of terms).

Final Steps

• Always simplify expressions by combining like terms after expanding.
• Pay attention to exponent rules when collecting like terms.

Conclusion

• Practice with examples similar to handouts.
• Ensure understanding of double distributive law and be cautious with signs and simplification.

Key Takeaways:

• Double distributive law and FOIL are essential tools in expanding binomials.
• Simplification involves careful combination of like terms.
• Practice is crucial for mastering these techniques.