Binomial Expansion and Coefficients - Revision Notes

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12.3Binomial Expansion and Coefficients

In these revision notes for Binomial Expansion and Coefficients, we cover the following key points:

  • What are binomial expressions?
  • What are the coefficients of binomial expressions?
  • How do we expand binomial expressions at small powers?
  • Why it is not suitable to use the FOIL Rule in higher power binomial expansions?
  • What is Pascal's Triangle? What is/are its shape and features?
  • What advantages and limitations does Pascal's Triangle have regarding the binomial coefficients?
  • What does the Binomial Coefficient Theorem say?
  • Why is Binomial Coefficient theorem more suitable to find the binomial coefficients of a binomial written in the expanded form?
  • How do we expand other types of binomials that are not expressed in the standard form?

Binomial Expansion and Coefficients Revision Notes

The term "binomial" refers to a polynomial containing two terms. When a binomial is raised to a certain power, it expands following specific rules. From previous tutorials, it is known that if we denote the terms of a binomial by a and b, the square of a binomial gives after expanding it

(a + b)2 = a2 + 2ab + b2

and for the cube of a binomial

(a + b)3 = a3 + 3a2 b + 3ab2 + b3

Expanding a binomial raised to a small power can be carried out in the long way by multiplying all terms in the resulting brackets with each other, but this procedure is very long and time-consuming. Therefore, it is necessary to find easier ways to expand such expressions. The main challenge to overcome in this respect consists of the value of the coefficients preceding each term after the expansion, as it is already clear that the general form of a binomial raised to the nth power is

(a + b)n = c1 ∙ an ∙ b0 + c2 ∙ an - 1 ∙ b1 + c3 ∙ an - 2 ∙ b2 + ⋯ + cn - 1 ∙ a1 ∙ bn - 1 + cn ∙ a0 ∙ bn

where c1, c2, c3, …, cn - 1 and cn are the coefficients preceding the corresponding terms of the given binomial after the expansion.

The first scientist who found a solution to this issue was Blaise Pascal, who proposed his famous triangle (Pascal's Triangle) for finding the binomial coefficients, which is a kind of equilateral triangle, where the lateral sides represent the coefficients preceding the highest degree terms an and bn.

We must consider the following things when dealing with the Pascal's Triangle:

  1. Each row represents the degree of the binomial starting from zero (in the uppermost row). The degree of the binomial increases by 1 each time we get to a lower row.
  2. The power of the first variable decreases by 1 when moving from left to right in the same row while the power of the second variable increases by 1 when moving from left to right in the same row.
  3. The sum of two adjacent numbers in a certain row give the coefficient of the term below them. For example, in the second row, we have two 1s. Therefore, the coefficient of the term below them is 1 + 1 = 2.
Math Tutorials: Binomial Expansion and Coefficients Example

The number of terms in the expanded form of a binomial is 1 more than the degree of the binomial.

Pascal's Triangle is very helpful in determining the binomial coefficients when dealing with small degree binomials. For higher degree ones, however, the use of Pascal's Triangle becomes more complicated, as the triangle's base widens. Therefore, it is necessary to use a more comprehensive method that is applicable in expressing the binomial coefficients of binomials of any degree. However, since this method includes some concepts from Combinatorics - an area of mathematics primarily concerned with counting, both as a mean and as a tool for obtaining results, and certain properties of finite structures - the first thing to do is to explain the meaning of these concepts before continuing with the general formula of the binomial coefficients.

In mathematics, the factorial of a number is a particular function that multiplies that number by all natural numbers that are smaller than it. The symbol of factorial is the exclamation mark (!). In general, we write for the factorial of any number n

n! = n ∙ (n - 1) ∙ (n - 2) ∙ … ∙ 3 ∙ 2 ∙ 1

As specials cases of factorial we have 1! = 1 and 0! = 1. On the other hand, the factorial of negative numbers does not exist.

In mathematics, a combination is a concept used to describe the number of ways a group of elements extracted from a set of items can be combined with each other. More precisely, a combination is a way of selecting items from a collection (without repetition) where the order of selection does not matter. We denote a combination by C(n, k), where n is the total number of available items and k is the number of items per group.

The general formula used to find the number of possible k-elements combinations in a set of n elements is

C(n,k) = n!/k!(n - k)!

We often express the (n, k) part not as a row but as a column instead. In this case, we don't write anymore the symbol C for combinations. In other word, the scripts below are equivalent.

C(n,k) ≡ (n/k)

The above form is used to express the Binomial Coefficients Theorem. This theorem, first discovered by Sir Isaac Newton, says that the coefficients preceding the variables in binomials raised to a given power are as follows:

(a + b)n = (n/0) ∙ an ∙ b0 + (n/1) ∙ an - 1 ∙ b1 + (n/2) ∙ an - 2 ∙ b2 + ⋯ + (n/k) ∙ an - k ∙ bk + ⋯ + (n/n - 1) ∙ a1 ∙ bn - 1 + (n/n) ∙ a0 ∙ bn

The general term of this binomial expression therefore is

(n/k) an - k bk

Hence, the algebraic form of expansion of the binomial expression (a + b)n is

(a + b)n = nk = 0(n/k) an - k bk

where the symbol "nk = 0" is an abbreviation that means "the sum of all terms from k = 0 to k = n".

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