Lecture Notes: FOIL in Binomial Expressions using Pascal's Triangle

Key Topics

• FOIL in binomial expressions
• Using Pascal's triangle
• Finding coefficients of terms

Methods to Expand Binomial Expressions

1. Multiplication

   • Multiply the binomial by itself the number of times indicated by the power.

2. Binomial Theorem

   • Use Pascal’s triangle to determine the coefficients for expansion.
   • Expand using the binomial theorem: ((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k})_

Pascal's Triangle

• Start with 1 at the top.
• Each row corresponds to a power of binomial expansion.
• Coefficients for expansion are found in the row corresponding to the binomial's exponent.
• Example row for exponent of 3: 1, 3, 3, 1.

Example: Expand (x - 2)^3

• Using Binomial Theorem:

  • Coefficients from Pascal’s triangle: 1, 3, 3, 1.
  • Resulting terms:
    • (1 \cdot x^3 \cdot (-2)^0 = x^3)
    • (3 \cdot x^2 \cdot (-2)^1 = -6x^2)
    • (3 \cdot x^1 \cdot (-2)^2 = 12x)
    • (1 \cdot x^0 \cdot (-2)^3 = -8)
  • Final expanded form: (x^3 - 6x^2 + 12x - 8)

• Confirming by FOILing: Multiply (x - 2) by itself three times to confirm the expansion.

Example: Expand (2x + 3y)^4

• Using Binomial Theorem:
  • Coefficients from Pascal’s triangle: 1, 4, 6, 4, 1.
  • Terms:
    • (2x) raised from 4 to 0.
    • (3y) raised from 0 to 4.
  • Coefficients involve products of powers of 2 and 3.
  • Example calculation for second term's coefficient: (4 \times 2^3 \times 3^1 = 96)

Finding Specific Terms

• Use combinations (nCr) to find coefficients.
• Formula: (nCr \times a^{n-r} \times b^r)
  • n is the exponent in the binomial.
  • r is the term order minus 1.

Example: Find 4th Term of (3x - 4y)^6

• Using Combination Formula:
  • Coefficient: 6C3
  • Terms: ((3x)^3) and ((-4y)^3)
  • Result: ((-34,560) x^3 y^3) with coefficient (-34,560)

Additional Practice

• Use the combination formula for given terms to directly compute specific terms without full expansion.

Important Tips

• Understand how to build Pascal’s triangle.
• Familiarize with nCr (combination) calculations to find specific coefficients.
• Practice both methods to confirm understanding.