A Guiding Principles about Binomial Theorem: Properties, Formulas and key Terms

The primary use of the binomial theorem is to assist in locating the expanded value of an algebraic expression that of the form (x + y)n. It is simple to determine the value of (x + y)2, (x + y)3, (a + b + c)2, and this may be accomplished by performing the algebraic operation of multiplying the value of the exponent by the appropriate number of times. However, in order to determine the expanded version of (x + y)17 or any other similar expressions with greater exponential values, a significant amount of maths is required. The binomial theorem is a tool that can be utilised to make the process simpler.

This expansion of the binomial theorem can have an exponent value that is either a fraction or a negative number. In this section, we will solely discuss non-negative values and their explanations.

Euclid, a well-known Greek mathematician who lived in the fourth century BC, is credited with making the earliest known mention of the binomial theorem. The binomial theorem is a mathematical principle that expresses the expansion of the algebraic expression (x + y)n as the sum of the terms that involve the individual exponents of the variables x and y. It states the principle for expanding the expression and expresses it as a sum. Each of the terms in a binomial expansion has a corresponding coefficient, which is a numeric value that is connected with it.

Binomial expansion:

The binomial theorem, also known as the binomial expansion, is a mathematical theorem that provides a formula for expanding the exponential power of a binomial expression. Another name for this theorem is the binomial expansion. Using the binomial theorem, the binomial expansion of the number (x + y)n can be written as follows:

   (x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + … + nCn-1 x1yn-1 + nCn x0yn

Binomial theorem formula:

When expanding any power of a binomial into the form of a series, the formula for the binomial theorem is utilised as part of the process. The formula for the binomial theorem is (a+b)n= nr=0 nCr an-r br, where n is an integer in the positive range, a and b are real numbers, and 0 is less than or equal to n. The binomial expressions such as (x + a)10, (2x + 5)3, (x – (1/x))4, and so on can be expanded with the use of this formula. The formula for the binomial theorem can be used to assist in expanding a binomial that has been raised to a particular power. Let’s start by getting a grasp of the formula for the binomial theorem and then moving on to its application in the following sections.

According to the binomial theorem, if x and y are both real numbers, then for any and all n∈N,

     (x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 + … + nCk xn-k yk +….+ nCn x0yn

⇒ (x + y)n = nk=0 nCk xn-k yk

where, nCr = n! / [r! (n – r)!]

Properties of binomial theorem:

• (n + 1) is the number of coefficients that can be found in the binomial expansion of the expression “(x + y)n.”

• The expansion of (x+y)n has n words plus one additional term.

• The initial and final terms, xn and yn, respectively, make up the sequence.

• After the expansion of (x + a)n has begun, the powers of x will drop from n all the way down to 0, while the powers of a will increase all the way from 0 all the way up to n.

• In the expansion of (x + y)n, the general term is the (r +1)th term, which can be represented as Tr+1, Tr+1 = nCr xn-r yr.

• Pascal’s triangle is an array that contains the binomial coefficients that are arranged in a manner that corresponds to the expansion. The formula for the binomial theorem encapsulates this pattern that emerged throughout time.

• The rth term from the end of the binomial expansion of (x + y)n is the same as the (n – r + 2)th term from the beginning of the expansion.

• If n is an even number, then the middle term in (x + y)n is equal to (n/2)+1. If n is an odd number, then the middle terms in (x + y)n are equal to (n+1)/2 and (n+3)/2.

Conclusion:

The Binomial Theorem states that the expansion takes more time and effort to calculate as the power of the expression increases. The Binomial Theorem makes it simple to compute the value of a binomial expression that has been multiplied by a very high power. The primary use of the binomial theorem is to assist in locating the expanded value of an algebraic expression of the form (x + y)n.The binomial theorem is a tool that can be utilised to make the process simpler.

The binomial theorem, also known as the binomial expansion, is a mathematical theorem that provides a formula for expanding the exponential power of a binomial expression.

The properties of binomial expansion are, 

The expansion of (x+y)n has n words plus one additional term. The initial and final terms, xn and yn, respectively, make up the sequence.

Pascal’s triangle is an array that contains the binomial coefficients that are arranged in a manner that corresponds to the expansion. The formula for the binomial theorem encapsulates this pattern that emerged throughout time.

The rth term from the end of the binomial expansion of (x + y)n is the same as the (n – r + 2)th term from the beginning of the expansion.