Glossary

12.4 Binomial Theorem

3 min read•june 25, 2024

and the 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 . 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

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  • 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](https://www.fiveableKeyTerm:(a+b)n)[(a+b)^n](https://www.fiveableKeyTerm:(a+b)^n)
    • nnth row represents coefficients of expansion of (a+b)n(a+b)^n
    • First number in nnth row is of ana^n, second number is coefficient of an−1ba^{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 aa and bb
  • Examples:
    • Third row of Pascal's Triangle (1, 3, 3, 1) corresponds to expansion of (a+b)3(a+b)^3
    • Expansion of (x+2)4(x+2)^4 using fifth row of Pascal's Triangle (1, 4, 6, 4, 1) is x4+8x3+24x2+32x+16x^4 + 8x^3 + 24x^2 + 32x + 16

Methods for binomial coefficients

  • Binomial coefficients are numbers in Pascal's Triangle denoted as (nk)\binom{n}{k} or C(n,k)C(n,k)
    • nn represents row number, kk represents position within row (starting from 0)
  • Binomial coefficients calculated using following methods:
    • Pascal's Triangle: Find number in nnth row and kkth position
    • Formula: (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}, where n!n! represents of nn
    • : (nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
  • Properties of binomial coefficients:
    • : (nk)=(nn−k)\binom{n}{k} = \binom{n}{n-k}
    • Sum of a row: Sum of numbers in nnth row of Pascal's Triangle equal to 2n2^n
  • Examples:
    • (52)\binom{5}{2} can be found in 5th row, 2nd position of Pascal's Triangle (10)
    • Using formula, (63)=6!3!(6−3)!=72036=20\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{720}{36} = 20
    • Using recursive relation, (74)=(63)+(64)=20+15=35\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(a+b)^n without directly multiplying terms
  • Binomial Theorem states: (a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
    • ∑\sum represents summation of terms from k=0k=0 to k=nk=n
    • (nk)\binom{n}{k} represents
    • an−ka^{n-k} and bkb^k represent powers of aa and bb in each term
  • To expand binomial using Binomial Theorem:
    1. Identify values of aa, bb, and nn
    2. Calculate binomial coefficients using one of methods mentioned earlier
    3. Substitute binomial coefficients and powers of aa and bb into Binomial Theorem formula
    4. Simplify resulting expression by combining like terms
  • Examples:
    • Expand (2x−3)3(2x-3)^3 using Binomial Theorem:
      • a=2xa=2x, b=−3b=-3, n=3n=3
      • Binomial coefficients: (30)=1\binom{3}{0}=1, (31)=3\binom{3}{1}=3, (32)=3\binom{3}{2}=3, (33)=1\binom{3}{3}=1
      • Expansion: (2x−3)3=1(2x)3+3(2x)2(−3)+3(2x)(−3)2+1(−3)3(2x-3)^3 = 1(2x)^3 + 3(2x)^2(-3) + 3(2x)(-3)^2 + 1(-3)^3
      • Simplified: 8x3−36x2+54x−278x^3 - 36x^2 + 54x - 27
    • Find coefficient of x5x^5 in expansion of (3x+2)7(3x+2)^7:
      • Coefficient of x5x^5 is (72)(3)5(2)2=21⋅243⋅4=20,412\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 of objects, which forms the mathematical foundation for the Binomial Theorem
  • : The Binomial Theorem is a specific case of polynomial expansion, providing a method for expanding (a+b)n(a+b)^n efficiently
  • : The Binomial Theorem can be viewed as an algebraic identity, expressing a complex expression in terms of simpler components
  • : In probability theory, the Binomial Theorem is used to calculate probabilities in situations with binary outcomes, forming the basis for the binomial distribution

Key Terms to Review (19)

(a+b)^n: The expression (a+b)^n, also known as the binomial expansion, represents the result of raising the sum of two variables, a and b, to the power of n. This expression is a fundamental concept in the Binomial Theorem, which provides a systematic way to expand and simplify such expressions.
Algebraic Identity: An algebraic identity is an equation that is true for all values of the variables involved. It is an equality that holds regardless of the specific numerical values substituted into the expression, unlike an algebraic equation which is only true for certain values of the variables.
Binomial Coefficient: The binomial coefficient, denoted as $\binom{n}{k}$, represents the number of ways to choose $k$ items from a set of $n$ items, without regard to order. It is a fundamental concept in combinatorics and plays a crucial role in the Binomial Theorem, which describes the expansion of binomial expressions.
Binomial Distribution: The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. It is a fundamental concept in probability theory and statistics, with applications in various fields.
Binomial Expansion Formula: The binomial expansion formula is a mathematical expression that allows for the expansion of a binomial expression raised to a power. It provides a systematic way to calculate the coefficients and terms in the expanded form of a binomial expression, enabling the simplification and evaluation of complex polynomial expressions.
Binomial Theorem: The Binomial Theorem is a mathematical 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 expansion, making it a powerful tool for working with polynomials and their properties.
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 mathematics.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number of times a variable appears in a term or an equation.
Combination: A combination is a way of selecting a subset of items from a larger set, where the order of the items in the subset does not matter. It is a fundamental concept in probability and combinatorics, and is closely related to the idea of permutations.
Combinatorics: Combinatorics is the branch of mathematics that deals with the study of discrete structures, including the counting, arrangement, and optimization of finite sets. It is a fundamental tool in various fields, including probability theory, statistics, and computer science.
Exponent: An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself. It is a concise way to express repeated multiplication of the same number.
Factorial: The factorial of a non-negative integer $n$ is the product of all positive integers less than or equal to $n$. It is denoted by $n!$ and represents the number of ways to arrange $n$ distinct objects in a sequence.
Isaac Newton: Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher who is widely regarded as one of the most influential scientists of all time. He is best known for his work on the laws of motion, the theory of gravitation, and his contributions to the development of calculus.
NCr: The term $nCr$ represents the number of ways to choose $r$ items from a set of $n$ items, without regard to order. It is a fundamental concept in the Binomial Theorem, which describes the expansion of binomial expressions.
Pascal's Triangle: Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. It is a fundamental concept in combinatorics and has applications in various areas of mathematics, including the Binomial Theorem.
Permutation: A permutation is an arrangement of a set of objects in a specific order. It represents the number of ways a group of items can be ordered or rearranged, where the order of the items matters.
Polynomial Expansion: Polynomial expansion is the process of multiplying a polynomial by another polynomial, resulting in a new polynomial with more terms. It is a fundamental operation in algebra and is particularly important in the context of the Binomial Theorem.
Recursive Relation: A recursive relation is a mathematical equation that defines a sequence or series of values, where each term in the sequence is determined by applying a formula to the preceding term(s) in the sequence. This type of relation is commonly used in the context of the Binomial Theorem to describe the patterns and relationships within the expansion of a binomial expression.
Symmetry: Symmetry is the property of an object or function that allows it to be transformed in some way, such as by rotation, reflection, or translation, without changing its overall appearance. This concept of invariance under transformation is a fundamental principle in mathematics, physics, and various other fields.
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