13.6 Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions like (a + b)^n. It provides a formula that makes expanding these expressions much easier, especially when dealing with large exponents.

Understanding the Binomial Theorem helps in various mathematical applications. From calculating probabilities to solving complex algebraic problems, this theorem is a fundamental concept that simplifies many mathematical operations.

Binomial Theorem and Its Applications

Application of Binomial Theorem

• Formula expands powers of binomials $(a + b)^n$ where $n$ is a non-negative integer
  • General form: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
    • $\binom{n}{k}$ binomial coefficient calculated using $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
      • $n!$ factorial of $n$ product of all positive integers less than or equal to $n$
• Expand binomial by substituting values of $a$, $b$, and $n$ into general form and simplify
  • Example expanding $(2x - 3)^4$:
    • $a = 2x$, $b = -3$, and $n = 4$ (where 4 is the exponent)
    • $(2x - 3)^4 = \sum_{k=0}^4 \binom{4}{k} (2x)^{4-k} (-3)^k$

Calculation of specific binomial terms

• Find specific term without expanding entire expression using general form and term's position
  • $(k+1)$th term in expansion of $(a + b)^n$ is $\binom{n}{k} a^{n-k} b^k$
• Determine coefficient by calculating binomial coefficient $\binom{n}{k}$ and multiplying by corresponding powers of $a$ and $b$
  • Example finding coefficient of $x^3$ term in expansion of $(2x - 3)^4$:
    • $x^3$ term corresponds to $k = 1$ (as $4-k = 3$)
    • $\binom{4}{1} = \frac{4!}{1!(4-1)!} = 4$
    • Coefficient of $x^3$ term: $4 \cdot (2x)^3 \cdot (-3)^1 = -96x^3$

Pascal's Triangle for binomial expansion

• Triangular array of numbers each number sum of two numbers directly above it
  • Numbers correspond to binomial coefficients $\binom{n}{k}$
    • $n$th row contains binomial coefficients for expansion of $(a + b)^n$
      • $k$th entry in $n$th row represents coefficient of term containing $a^{n-k} b^k$
• Expand binomial using Pascal's Triangle by writing out coefficients from corresponding row and multiplying each coefficient by appropriate powers of $a$ and $b$
  • Example expanding $(a + b)^3$ using Pascal's Triangle:
    • 3rd row of Pascal's Triangle: 1 3 3 1
    • $(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3$

Related Concepts in Algebraic Expansion

• Permutation: Used in calculating binomial coefficients, represents number of ways to arrange n distinct objects
• Polynomial: A more general form of expression that includes binomials, can be expanded using similar techniques
• Algebraic expansion: The process of multiplying out brackets in an expression, which is what the Binomial Theorem helps accomplish efficiently