12.4 Binomial Theorem

Pascal's Triangle and the Binomial Theorem are powerful tools for expanding binomial expressions. They provide a visual and mathematical way to find coefficients and simplify complex calculations in algebra and probability.

These concepts are crucial for understanding polynomial expansions and combinatorics. Mastering them will help you tackle more advanced topics in algebra and lay a solid foundation for future math courses.

Pascal's Triangle and the Binomial Theorem

Construction of Pascal's Triangle

• Triangular array of numbers each number is the sum of the two numbers directly above it
  • First and last numbers in each row always 1
  • To find any number inside the triangle add the two numbers directly above it
• Rows of Pascal's Triangle correspond to powers of a binomial expansion $(a+b)^n$
  • $n$th row represents coefficients of expansion of $(a+b)^n$
  • First number in $n$th row is coefficient of $a^n$, second number is coefficient of $a^{n-1}b$, and so on
• To expand binomial expression using Pascal's Triangle:
  1. Identify row corresponding to power of binomial
  2. Write out terms of expansion using numbers in row as coefficients
  3. Multiply each coefficient by appropriate powers of $a$ and $b$
• Examples:
  • Third row of Pascal's Triangle (1, 3, 3, 1) corresponds to expansion of $(a+b)^3$
  • Expansion of $(x+2)^4$ using fifth row of Pascal's Triangle (1, 4, 6, 4, 1) is $x^4 + 8x^3 + 24x^2 + 32x + 16$

Methods for binomial coefficients

• Binomial coefficients are numbers in Pascal's Triangle denoted as $\binom{n}{k}$ or $C(n,k)$
  • $n$ represents row number, $k$ represents position within row (starting from 0)
• Binomial coefficients calculated using following methods:
  • Pascal's Triangle: Find number in $n$th row and $k$th position
  • Formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ represents factorial of $n$
  • Recursive relation: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
• Properties of binomial coefficients:
  • Symmetry: $\binom{n}{k} = \binom{n}{n-k}$
  • Sum of a row: Sum of numbers in $n$th row of Pascal's Triangle equal to $2^n$
• Examples:
  • $\binom{5}{2}$ can be found in 5th row, 2nd position of Pascal's Triangle (10)
  • Using formula, $\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{720}{36} = 20$
  • Using recursive relation, $\binom{7}{4} = \binom{6}{3} + \binom{6}{4} = 20 + 15 = 35$

Applications of Binomial Theorem

• Binomial Theorem is formula for expanding powers of binomials $(a+b)^n$ without directly multiplying terms
• Binomial Theorem states: $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
  • $\sum$ represents summation of terms from $k=0$ to $k=n$
  • $\binom{n}{k}$ represents binomial coefficient
  • $a^{n-k}$ and $b^k$ represent powers of $a$ and $b$ in each term
• To expand binomial using Binomial Theorem:
  1. Identify values of $a$, $b$, and $n$
  2. Calculate binomial coefficients using one of methods mentioned earlier
  3. Substitute binomial coefficients and powers of $a$ and $b$ into Binomial Theorem formula
  4. Simplify resulting expression by combining like terms
• Examples:
  • Expand $(2x-3)^3$ using Binomial Theorem:
    • $a=2x$, $b=-3$, $n=3$
    • Binomial coefficients: $\binom{3}{0}=1$, $\binom{3}{1}=3$, $\binom{3}{2}=3$, $\binom{3}{3}=1$
    • Expansion: $(2x-3)^3 = 1(2x)^3 + 3(2x)^2(-3) + 3(2x)(-3)^2 + 1(-3)^3$
    • Simplified: $8x^3 - 36x^2 + 54x - 27$
  • Find coefficient of $x^5$ in expansion of $(3x+2)^7$:
    • Coefficient of $x^5$ is $\binom{7}{2}(3)^5(2)^2 = 21 \cdot 243 \cdot 4 = 20,412$

Additional Concepts and Applications

• Combinatorics: The study of counting, arrangement, and combination of objects, which forms the mathematical foundation for the Binomial Theorem
• Polynomial expansion: The Binomial Theorem is a specific case of polynomial expansion, providing a method for expanding $(a+b)^n$ efficiently
• Algebraic identity: The Binomial Theorem can be viewed as an algebraic identity, expressing a complex expression in terms of simpler components
• Binomial distribution: In probability theory, the Binomial Theorem is used to calculate probabilities in situations with binary outcomes, forming the basis for the binomial distribution