Statement and Proof of the Binomial Theorem

For the most part, the binomial theorem is useful in determining the expanded value of an algebraic expression of the type (x + y)n. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is straightforward and may be accomplished by algebraically multiplying the exponent value by the number of times it appears in the equation. However, calculating the expanded form of (x + y)17 or other similar expressions with larger exponential values necessitates a significant amount of computation. Using the binomial theorem, it is possible to make things a little easier. 

When applying this binomial theorem expansion, the exponent value might be either a negative number or a fraction. We will restrict our explanations to solely non-negative values in this section.

What is a binomial theorem? 

The binomial theorem was first mentioned in the 4th century BC by a great Greek mathematician by the name of Euclids, who lived during the time of the ancient Greeks. The binomial theorem expresses the principle for expanding the algebraic expression (x + y)n and expressing it as a sum of the terms involving individual exponents of the variables x and y. It is derived from the binomial theorem and can be found in the literature. Whenever a phrase appears in a binomial expansion, it is connected with a numerical value, which is referred to as the coefficient.

Statement:  

A non-negative power of binomial (x + y) can be expanded into a sum of the form, according to the binomial theorem. 

 (x+y)n = nC0 xny0 + nC1 xn-1y1 + nC2 xn-2 y2 and so on up to and including the sum of the first nCn-1 x1yn-1+nCnx0yn.

 in which the positive integer nCk is known as a binomial coefficient, and where the number n=>0 is an integer.  

Please keep in mind that when an exponent is 0, the corresponding power expression is 1. Because this multiplication component is frequently omitted from the term, the right-hand side is frequently written directly as nC0 xn +…. This formula is sometimes known as the binomial formula or the binomial identity, depending on the context.

Binomial theorem expansion proof:

Assume that x, a, and n∈N. Let us use the principle of mathematical induction to show the binomial theorem formula in its entirety. It is sufficient to demonstrate for n = 1, n = 2, for n = k≥2, and for n = k+ 1 that the condition is met. 

It should go without saying that (x + y)1 = x + y and

 (x + y)2 = (x + y)(x + y)  

x2 + xy + xy + y2 (using distributive property)

X2 +2xy + y2

As a result, the conclusion holds for both n = 1 and n = 2. Assume that k is a positive integer. Let us demonstrate that the conclusion is correct for k ≥ 2. 

Assuming (x + y)n = ∑nr=0nCr xn-ryr

 (x + y)k = ∑kr=0kCr xk-r yr

 ⇒ (x+y)k = kC0 xky0 + 1C1 xk-1y1 + kC2 xk-2 y2 + … + kCr xk-r yr +….+ kCk x0 yk 

⇒ (x+y)k = xk + kC1 xk-1 y1 + kC2 xk-2 y2 + … + kCr xy-r yr +….+ yk

 As a result, the conclusion is correct for n = k≥2.

 Now evaluate the expansion for n = k + 1 as a function of time.

 (x + y)k+1 = (x + y) (x + y)k 

= (x + y) (xk + kC1 xk-1 y1 + kC2 xk-2 y2 + … + kCr xk-r yr +….+ yk) 

= xk+1 + (1 + kC1)xky + (kC1 + kC2) xk-1 y2 + … + (kCr-1 + kCr) xk-r+1 yr + … + (kCk-1 + 1) xyk + yk+1

 = xk+1 + k+1C1xky + k+1C2 xk-1 y2 + … + k+1Cr xk-r+1 yr + … + k+1Ck x yk + yk+1 [Because nCr + nCr-1 = n+1 Cr].

 As a result, the conclusion is valid for n = k+1. This result is true for all positive integers ‘n’, as demonstrated through mathematical induction. As a result, it was proven.

Conclusion:

An equation that has been raised to any finite power can be expanded using the Binomial Theorem, which is the method of doing so. A binomial Theorem is a strong instrument of extension that can be used in a variety of fields such as algebra, probability, and statistics.

For the most part, the binomial theorem is useful in determining the expanded value of an algebraic expression of the type (x + y)n. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is straightforward and may be accomplished by algebraically multiplying the exponent value by the number of times it appears in the equation. When applying this binomial theorem expansion, the exponent value might be either a negative number or a fraction.

A non-negative power of binomial (x + y) can be expanded into a sum of the form, according to the binomial theorem.

 (x+y)n = nC0 xn y0 + nC1 xn-1 y1 + nC2 xn-2 y2 and so on up to and including the sum of the first nCn-1 x1 yn-1 +nCnx0yn.