Binomial expansion

A binomial expression can be amplified by using the binomial theorem which is an algebraic method. It shows what happens when a binomial is multiplied by itself (as many times as you want it to). For example, take the expression (4x + y)7. It would take an extremely long time to multiply the binomial (4x+y) out seven times. This theorem provides a shortcut or a formula that tends to expand the form of this expression. As per the theorem, it is possible to expand the power (x+y)n into a sum involving terms of the form a(x)byc, where the exponents b and c are non-negative integers with b+c=n and the coefficient a of each term is a specific positive integer relying on n and b. When an exponent is zero, the corresponding power is usually omitted from the term (so that 3x2yo would be written as 3x2). 

The general term of binomial theorem

The binomial expansion is a method that is used to expand the binomials with power in an algebraic expression. In algebra, a binomial is considered as an algebraic expression with exactly two terms (the prefix ‘bi’ refers to the number 2). If a binomial expression (x + y)n

has to be expanded then the binomial expansion formula can be utilised to express it in simple expression terms in the form of ax+by+c in which ‘b’ and ‘c’ are non-negative integers. Note that the value of ‘a’ is completely reliant on the value of ‘n’ and ‘b’. 

Properties of binomial coefficients

• There are n+1 terms in total in binomial expansion
•  yn is the last term and xn is the initial term . 
•  As we progress from the first to the last phrase, the exponents of x decrease by one. The exponents of y on the other hand by one while the exponents of x decrease by one. Furthermore, the sum of both exponents in each term equals n. 
• The coefficient of the following phrase is calculated very easily by multiplying the coefficient of each term by the exponents of x in that term and then dividing the result by the number of terms.

Formula for binomial theorem

The formula for binomial theorem is written as:

(x+y)n=∑k=r=0r=n(ncr)xn-kyk

Another thing to note is that ‘n’ is factorial notation. It reflects the product of the whole number starting between 1 and n. 

The following are some examples of binomial expansion. 

(x+y)1=x+y

(x+y)2=x²+2xy+y²

(x+y)3=x³+3x²y+3xy²+y³

General term of binomial expansion (mathematical form) 

The binomial expansion of any form (a + x) can be increased when it is raised to any power, say ‘n’ using the binomial expansion formula mentioned below: 

( a + x )n = an + nan-1x +(n(n−1)/2)an-2 x2…… + xn

The expansion always has (n+ 1) terms. The general term of binomial expansion can also be written as:

(a+x)n=∑k=(ncr)xn-kyk

Note that the factorial is given by

N! = 1 . 2 . 3 … n

0! = 1

Important terms involved in binomial expansion

The binomial expansion elevated to a certain power is given by binomial theorem. It is also known as binomial expansion. Several often used terminology in the expansion are listed below:

• General term 
• Middle term 
• Independent term 
• Numerically greatest term 
• To determine a particular term
• The ratio of consecutive terms also known as the coefficients

Usage of binomial expansion formula

Various mathematical and scientific calculations are done using the formula of binomial expansion. Some of them are stated below:

• Kinematic and gravitational time dilation 
• Kinetic energy 
• Electric quadrupole pole
• Determining the relativity factor gamma 

Conclusion

The binomial theorem is a quick way to multiply or expand a binomial statement. The intensity of the expressiveness has been amplified significantly. The addition of similar statements is always difficult with large powers and expressions, as we all know. Still, binomial expansions and formulas are extremely helpful in this area. It’s used in all Mathematical and scientific computations that involve these types of equations. Limited concepts in Physics that use the binomial expansion formula relatively frequently are

Kinematic and gravitational time dilation, Kinetic energy, Electric quadrupole pole, and Determining the relativity factor gamma.