Glossary

13.6 Binomial Theorem

2 min read•june 24, 2024

The 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

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  • Formula expands powers of binomials (a+b)n(a + b)^n where nn is a non-negative integer
    • General form: (a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
      • (nk)\binom{n}{k} calculated using (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}
        • n!n! of nn product of all positive integers less than or equal to nn
  • Expand binomial by substituting values of aa, bb, and nn into general form and simplify
    • Example expanding (2x−3)4(2x - 3)^4:
      • a=2xa = 2x, b=−3b = -3, and n=4n = 4 (where 4 is the )
      • (2x−3)4=∑k=04(4k)(2x)4−k(−3)k(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)(k+1)th term in of (a+b)n(a + b)^n is (nk)an−kbk\binom{n}{k} a^{n-k} b^k
  • Determine coefficient by calculating binomial coefficient (nk)\binom{n}{k} and multiplying by corresponding powers of aa and bb
    • Example finding coefficient of x3x^3 term in expansion of (2x−3)4(2x - 3)^4:
      • x3x^3 term corresponds to k=1k = 1 (as 4−k=34-k = 3)
      • (41)=4!1!(4−1)!=4\binom{4}{1} = \frac{4!}{1!(4-1)!} = 4
      • Coefficient of x3x^3 term: 4⋅(2x)3⋅(−3)1=−96x34 \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 (nk)\binom{n}{k}
      • nnth row contains binomial coefficients for expansion of (a+b)n(a + b)^n
        • kkth entry in nnth row represents coefficient of term containing an−kbka^{n-k} b^k
  • Expand binomial using by writing out coefficients from corresponding row and multiplying each coefficient by appropriate powers of aa and bb
    • Example expanding (a+b)3(a + b)^3 using Pascal's Triangle:
      • 3rd row of Pascal's Triangle: 1 3 3 1
      • (a+b)3=1a3+3a2b+3ab2+1b3(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3
  • : Used in calculating binomial coefficients, represents number of ways to arrange n distinct objects
  • : A more general form of expression that includes binomials, can be expanded using similar techniques
  • : The process of multiplying out brackets in an expression, which is what the Binomial Theorem helps accomplish efficiently

Key Terms to Review (21)

(x + y)^n: The expression (x + y)^n represents the binomial expansion, which is the result of raising the sum of two variables, x and y, to a power n. This term is central to the Binomial Theorem, a powerful mathematical tool used to expand and simplify binomial expressions.
Algebraic Expansion: Algebraic expansion is the process of multiplying or raising algebraic expressions to a power in order to create a new expression with more terms. It involves applying the distributive property to combine like terms and simplify the resulting expression.
Binomial Coefficient: The binomial coefficient is a mathematical concept that represents the number of ways to choose a certain number of items from a set, without regard to order. It is a fundamental principle in combinatorics and has applications in various areas of mathematics, including probability, statistics, and the binomial theorem.
Binomial Expansion Formula: The binomial expansion formula is a mathematical expression used to expand a binomial raised to a power. It provides a systematic way to calculate the coefficients and terms in the expanded expression, allowing for the representation of a power of a sum or difference as a polynomial.
Binomial Theorem: The Binomial Theorem is a formula that allows for the expansion of binomial expressions raised to a power. It provides a systematic way to calculate the coefficients and exponents of the terms in the expanded form of a binomial expression.
Blaise Pascal: Blaise Pascal was a renowned French mathematician, physicist, inventor, and philosopher who made significant contributions to various fields, including the development of the binomial theorem, a fundamental concept in college algebra.
Combination: A combination is a way of selecting a set of items from a larger group, where the order of the items does not matter. Combinations are a fundamental concept in the fields of mathematics, probability, and the binomial theorem.
Combinations: A combination is a selection of items from a larger pool where the order does not matter. It is calculated using the binomial coefficient formula.
Expansion: Expansion refers to the process of increasing the size, scope, or scale of something. In the context of the Binomial Theorem, expansion describes the mathematical operation of expanding a binomial expression into a sum of terms, revealing the individual components and their respective coefficients.
Exponent: An exponent indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents the power to which a number or variable is raised, and it is a fundamental concept in algebra, exponential functions, logarithmic functions, and other areas of mathematics.
Factorial: A factorial, denoted by $n!$, is the product of all positive integers less than or equal to $n$. It is used in permutations, combinations, and other mathematical calculations.
Factorial: The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. It is a fundamental concept in combinatorics and probability that has applications in various areas of mathematics, including the Binomial Theorem and Counting Principles.
Increasing function: An increasing function is a function where the value of the output increases as the input increases. Mathematically, for any two values $x_1$ and $x_2$ such that $x_1 < x_2$, the function $f(x)$ satisfies $f(x_1) \leq f(x_2)$.
Isaac Newton: Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher who is widely recognized as one of the most influential scientists of all time. His groundbreaking work in the areas of classical mechanics, optics, and calculus laid the foundation for many modern scientific concepts and principles.
NCr: nCr, also known as the binomial coefficient, is a fundamental concept in combinatorics that represents the number of ways to choose r items from a set of n items, without regard to order. It is a crucial term in understanding the Binomial Theorem and various counting principles.
Pascal's Triangle: Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. It is a fundamental concept in combinatorics and has numerous applications in mathematics, particularly in the study of binomial coefficients and probability theory.
Permutation: A permutation is an arrangement of all or part of a set of objects in a specific order. The order of the elements is crucial, and changing the order produces a different permutation.
Permutation: A permutation is an ordered arrangement of a set of objects or elements. It refers to the different ways in which a group of items can be arranged or ordered, taking into account the order of the elements. Permutations are fundamental concepts in the fields of combinatorics, probability, and the Binomial Theorem.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
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