Lecture Notes: Expanding Binomial Expressions Using Pascal's Triangle

Overview

• Focus on using Pascal's Triangle to foil binomial expressions and finding specific term coefficients.
• Example expressions include ((x - 2)^3) and ((2x + 3y)^4).

Methods for Expanding Binomials

1. Foiling by Multiplication
   • Multiply the binomial expression several times.
2. Using the Binomial Theorem
   • Employ Pascal's Triangle for efficient expansion.

Pascal’s Triangle and Coefficients

• Construct Pascal’s Triangle.
  • Example: For exponent 3, use coefficients 1, 3, 3, 1.
  • Each row corresponds to the powers in the expansion.
• Coefficients for expressions are drawn from the corresponding row.

Example 1: ((x - 2)^3)

Using Binomial Theorem

• Coefficients: 1, 3, 3, 1 from Pascal’s Triangle.
• Expand:
  • First term: (x^3)
  • Second term: (-6x^2)
  • Third term: (12x)
  • Fourth term: (-8)

Verification by Foiling

• Foil the expression by multiplying directly.
• Combine like terms to verify.

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

Using Binomial Theorem

• Coefficients: 1, 4, 6, 4, 1.
• Expand:
  • First term: (16x^4)
  • Second term: (96x^3y)
  • Third term: (216x^2y^2)
  • Fourth term: (216xy^3)
  • Fifth term: (81y^4)

Finding Specific Terms

• Example: Coefficient of the fourth term in ((3x - 4y)^6).
  • Use Pascal's Triangle for coefficients.
  • Calculate using: (nCr \times a^{n-r} \times b^r).
  • Combine values to find the specific coefficient.

Combinations and Pascal's Triangle

• Use combinations (C(n, r)) to find specific coefficients in Pascal's Triangle.
• Formula: (nCr = \frac{n!}{(n-r)!r!}).

Formula for Finding Terms Directly

• Expression: (a + b)^n) to find any term:
  • Formula: (nCr \times a^{n-r} \times b^r).
  • R is the term number minus one.

Practice Problems

• Find the fourth term and its coefficient for ((3x - 4y)^6).
• Confirm using the binomial formula.

Conclusion

• Binomial theorem and Pascal's Triangle simplify expansion of binomials.
• Useful for both manual computation and verification of results.