Binomial Theorem- Formula, Expansion

The expanded value of an algebraic expression of the form (x + y)n is found using the binomial theorem. Finding the value of (x + y)2, (x + y)3, and (a + b + c)2 is simple and may be done simply by multiplying the exponent value by the number of times. However, calculating the expanded form of (x + y)17 or other formulas with greater exponential values takes too long. With the help of the binomial theorem, it can be made easy. This binomial theorem expansion’s exponent value can be a fraction or a negative number.

Statement

According to the binomial theorem, any non-negative power of binomial (x + y) can be expanded into a total of the form,

(x+y)n = nC0 xny0 + nC1 xn-1 y1 + nC2 xn-2 y2 +… + nCn-1 x1yn-1 + nCn x0yn-1 + nCn x0 yn-1. 

Each nCk is a positive integer known as a binomial coefficient, and n≥0 is an integer.

When an exponent is 0, the power expression corresponding to it is 1. Because the multiplicative component is frequently omitted from the term, the right-hand side is frequently expressed as nC0 xn +…. This formula is also known as the binomial identity or binomial formula. The binomial theorem can be written as, using summation notation.

(x+y)n = ∑nk=0nCk xn-k yk =∑nk=0 nCk xk yn-k

Binomial expansion

The binomial theorem, commonly known as the binomial expansion, gives the formula for expanding a binomial expression’s exponential power. The following is the binomial expansion of (x + y)n using the binomial theorem:

(x+y)n = nC0 xn y0 + nC1 xn-1 y1 + nC2 xn-2 y2 + … + nCn-1 x1 yn-1 + nCn x0 yn. 

Binomial theorem formula

In order to expand any binomial power into a series, the binomial theorem formula is needed. (a+b)n = ∑nr=0 nCr an-r br, where n is a positive integer, a, b are real integers, and 0<r≤n is the binomial theorem formula. This formula can be used to expand binomial statements like (x + a)10, (2x + 5)3, (x – (1/x))4, and others. The formula for the binomial theorem aids in the expansion of a binomial raised to a particular power. In the following sections, we’ll look at the binomial theorem formula and how it’s used.

If x and y are real numbers, then for any n∈N, the binomial theorem holds.

(x+y)n = nC0 xny0 + nC1 xn-1 y1 + nC2 xn-2 y2 + … + nCk xn-k yk +….+ nCn x0 yn

⇒ (x + y)n = ∑nk=0 nCkxn-k yk

where, nCr = n! / [r! (n – r)!]

Important terms of the binomial theorem

The terms related to binomial expansion using the binomial theorem are listed below to help you find them. The following are the specifics of each of the terms.

General term

This word represents all of the terms in the (x + y)n binomial expansion. Tr+1 = nCr xn-r yr is the general term in the binomial expansion of (x + y)n. The r-value, in this case, is one less than the number of binomial expansion terms. The coefficient is also nCr, and the sum of the exponents of the variables x and y is n.

Middle term

The total number of terms in the (x + y)n expansion equals n + 1. The binomial expansion’s middle term is determined by the value of n. The number of middle terms and the value of n are determined by whether n is even or odd. There is just one middle term for an even value of n, and it is the (n/2 + 1)th term. There are two middle terms for an odd number of n, and the two middle terms are n/2 and n/2 + 1.

Conclusion

Binomial Theorem – As the power increases, calculating the expansion gets more difficult. The Binomial Theorem can be used to calculate a binomial statement that has been raised to very big power. The expanded value of an algebraic expression of the form (x + y)n is found using the binomial theorem. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is simple and maybe done simply by multiplying the exponent value by the number of times.

When an exponent is 0, the power expression corresponding to it is 1. Because the multiplicative component is frequently omitted from the term, the right-hand side is frequently expressed as nC0 xn +…. This formula is also known as the binomial identity or binomial formula.

The binomial theorem, commonly known as the binomial expansion, gives the formula for expanding a binomial expression’s exponential power. This word represents all of the terms in the (x + y)n binomial expansion. Tr+1 = nCr xn-r yr is the general term in the binomial expansion of (x + y)n.