Math 100: Special Products (Section 5-3)

Review of FOIL Method

• FOIL is used to multiply two binomials.
  • F: First terms
  • O: Outside terms
  • I: Inside terms
  • L: Last terms
• FOIL is essentially distributing:
  • Example: ( (x + 5)(x + 6) )
    • First: ( x \times x = x^2 )
    • Outside: ( x \times 6 = 6x )
    • Inside: ( 5 \times x = 5x )
    • Last: ( 5 \times 6 = 30 )
    • Combine like terms: ( x^2 + 11x + 30 )

Special Products

• Multiplying binomials that are almost identical, differing only by addition and subtraction:
  • Example: ( (a + b)(a - b) )
    • ( a \times a = a^2 )
    • ( a \times -b = -ab )
    • ( b \times a = ab )
    • ( b \times -b = -b^2 )
    • Middle terms cancel: Result is ( a^2 - b^2 )

Example with Numbers

• Multiply: ( (4y + 3)(4y - 3) )
  • ( 4y \times 4y = 16y^2 )
  • ( 4y \times -3 = -12y )
  • ( 3 \times 4y = 12y )
  • ( 3 \times -3 = -9 )
  • Middle terms cancel: Result is ( 16y^2 - 9 )

Squaring a Binomial

• Common mistake: ( (a + b)^2 \neq a^2 + b^2 )

• Correct approach: ( (a + b)(a + b) )

  • ( a \times a = a^2 )
  • ( a \times b = ab )
  • ( b \times a = ab )
  • ( b \times b = b^2 )
  • Combine like terms: ( a^2 + 2ab + b^2 )

• Example with subtraction: ( (x - 4)^2 )

  • ( x \times x = x^2 )
  • ( x \times -4 = -4x )
  • ( -4 \times x = -4x )
  • ( -4 \times -4 = 16 )
  • Combine like terms: ( x^2 - 8x + 16 )

Key Takeaways

• Important to fully distribute and not skip steps.
• Understanding FOIL and distribution is more beneficial than memorizing shortcuts.
• Ensure to write expressions correctly to avoid common mistakes.
• Looks at section 5-4 in the next lesson.