Binomial theorem

When you are given a set of numbers, you can operate on them easily like adding them, multiplying them, and so on. However, when you have to deal with ‘n’ numbers of elements, finding the result is very difficult due to the lack of knowledge about the values. Therefore, some kind of statistical analysis needs to be done. A lot of ways are used for such a kind of mathematical and statistical calculations. But out of all these techniques, it is the binomial theorem that has proven to be more accurate and precise. This theorem is used for operating on different types of number series like 1, 2, 3, 4, 5, and so on up to a n.

What is a binomial theorem?

A binomial expression is described as (x+y) because it has two different terms x and y. When we consider (x+y)1, the resultant is (x+y). For (x+y)2, the expansion can be written as x2+2xy+y2. Getting the right value of all these known expansions is easy but not when you have an expression of (x+y)n where n can be anything like an integer, a decimal, or a fraction.

To help in solving such problems, the binomial theorem is used where the expression is stated as:

It is possible to expand any non-negative power of a binomial (x+y) into a sum of the form as:

(x+y)n= 0nxny0+ 1nxn-1y1+ 2nxn-2y2+⋯+ n-1nx1yn-1+ nnx0yn

Mathematically, if we consider n that belongs to the natural number set, two independent variables like a and b that belong to the real number set, the general formula for the binomial expansion will be:

(x+y)n=∑r=0n  nCrxn-ryr=∑r=0n Tr+1

where nCr (=Cr ) can be defined as n!/(n-r)!r!

Let’s take an example of (x+y)5 + (x-y)5. By applying the binomial theorem, this algebraic expression can be expanded in the following manner:

(x+y)5 + (x-y)5= 2 . [5C0x5 + 5C2x3y2 + 5C4xy4]

= 2(x5 + 10x3y2 + 5xy4)

Now, if x = √3 and y = 1, then the expression will have a final value of:

(√3 + 1)5 + (√3 – 1)5 = 2[(√3)5 + 10(√3)3 + 5(√3)]

= 88√3

If we consider two variables, x and y, then the binomial expansion from index 0 to 8 can be expressed as:

(x+y)0=1(x+y)2=x+y(x+y)2=x2+2xy+y2(x+y)3=x3+3x2y+3xy2+y3(x+y)4=x4+4x3y+6x2y2+4xy3+y4(x+y)3=x5+5x4y+10x3y2+10x2y3+5xy4+y5(x+y)5=x5+6x5y+15x4y2+20x3y3+15x2y4+6xy5+y6(x+y)7=x7+7x6y+21x5y2+35x4y3+35x3y4+21x2y5+7xy6+y7(x+y)5=x8+8x7y+26x6y2+56x5y3+70x4y4+56x3y5+28x2y6+8xy7+y8.

It is possible to apply a binomial theorem for negative variables. But the indices cannot be negative because as per the expansion formula, the index value will indicate the upper limit of the sum of the series that can’t be negative.

What are binomial coefficients?

The coefficients in the binomial theorem can be defined as the integral value attached with the algebraic variables. For example, let’s say that the third element of an expansion series is 3C1a. Now, this term can be considered as the combined value that will be evaluated as:

3C1a = a  [3! / (3-1)!1!]

= (3.2.1.a)/(2!)

= 6a/2

= 3a

In every binomial expansion, there are a lot of terms involved and connected by the (+) operator which are considered as coefficients. For example:

nC0+ nC1+ nC2+…+ nCn=2 n

Similarly, if we consider the series of even and odd coefficients, the binomial theorem expressions can be written in the following manner:

nC0+ nC2+ nC4+…=2n-1 nC1+ nC3+ nC5+…=2n-1

Similar result,

C1 + C3 + C5 + C7 + C9 + C11 + C13 + C15 + C17 + ………. + C2n+1 = 2n

The properties of binomial coefficients can be considered as the following:

1. When the binomial coefficients are added together, their total sum is considered as 2n.
2. If the binomial coefficients are arranged with alternate plus and minus operators, the resultant value of the series will be 0.

 3. When the squares of the binomial coefficients are added together, the total sum is given by [(2n)! / (n!)2].

i) The sum of even terms and odd terms in a binomial series will be the same as 2n-1.

ii) For a binomial expansion series like C1+2C2+3C3…..+nCn, the resultant value will be n.2n-1.

Terms used in the binomial expansion

There are three main terms in the binomial expansion that need to be known for evaluating any indexed binomial or calculating the coefficient orders and sums. They are:

1. The General term is always denoted by using an alphabetical expression as its value can change with the coefficient, its place in the series, and even the type of binomial expansion itself. It is often denoted by Ti+1 which can also be written as nCi . Xn-I . Yi.
2. Middle Term
   The middle term in the expansion of (x+y)n depends upon the value of n.

(i) If n is even
Total number of terms in the expansion =n+1 (odd)
There is only one middle term n2+1th

(ii) If n is odd
Total number of terms in the expansion =n+1 (even)
There is two middle terms, n+12th   and n+12+1th .

1. The greatest term can be written in the form of a complex formula where the mod operator is also used.
2. A term not having a particular variable, say x, is known as the independent term. So, for a binomial expression like [axp+(b/xq)]n, the independent term of x will be given as:

Tr+1 = nCran-r.br

Where, r = [np/(p+q)]

Let’s consider two consecutive terms of a binomial expression: xr and xr+1. The coefficient of xr is nCr-1 and that of xr+1 is nCr. Therefore, the ratio of these two consecutive terms or coefficients will be:

(nCr/nCr-1) = (n-r+1)/r

In what ways can binomial theorems be used?

The binomial theorem has extensive use in the mathematics and instrumental field along with science. In this below section, a few such applications will be discussed to have a better idea about how this binomial expression is used.

1. It will become possible for anyone to find a remainder of numeric expressions like 1212/9 or 5556/4789. One can write these kinds of expressions in the form of binomial and apply the theorem properties to get the right answer.
2. When you have to find a value of xn where x and n have higher values like above 20, using the binomial theorem will be the best option; it will quickly give you the shortest possible answer with maximum accuracy and precision.
3. Sometimes, you will often see that mathematics papers have problems like “Show that 129 + 912 can be divided by 15”. To solve these cases, the only option you will have is to use the binomial expression.
4. AP, GP, and HP are derived from this binomial theorem only because they too consider a series of infinite lengths where the last number is represented by n.

Apart from these practical uses, the binomial theorem is also the basis for the concept of mathematical induction, advanced algebra, logarithm, permutation and combination, and even statistics.

Conclusion

Now that you have learnt everything about binomial theorems, it will become much easier for you to come up with the right solutions for any kind of complicated expressions where high values of indices are involved. Moreover, binomial coefficients are based on combination formulas which is why you can easily calculate the values while using the factorial concepts. As this mathematical concept is applicable for a wide number of scenarios, learning about them properly is extremely important. Moreover, quite a lot of questions based on binomial expansion come in different types of entrance exams, especially IIT-JEE mains and advanced.