Expansion of Binomial Expressions

Introduction

Before heading to the binomial expressions, it is important for you to understand what binomial exactly means. A binomial is a two-term algebraic expression (i.e. two variables expression). Binomials include expressions like a+b, x+y, and so on. 

When a binomial is increased to exponents 2 and 3, we have a series of algebraic identities to find the expansion. 

For example: 

(a+b)2=a2+2ab+b2

But what happens if the exponents are larger? Finding the expansion manually is time-consuming. This is made easier by using the binomial expansion formula. 

About Binomial Theorem

A Binomial Theorem is a useful expansion technique that can be used in fields like probability, algebra, and more. 

It is an algebraic expression that has two terms that are not the same. For example, a + b, a3+b3, and so on.

Binomial Theorem: Let n ∈ N,x,y,∈ R then

(x+y)n will be equal to r=0nncrxn-ryr where,

ncr= n!(n-r)! r!

What Are Binomial Expansion Formulas?

The binomial expansion formula is used to find the powers of binomials that cannot be expanded using algebraic identities, as we covered in the previous section.

The binomial theorem is another name for the binomial expansion formula.

For Natural Powers

The expansion of (x+y)n, where ‘n’ is a natural number, is given by this binomial expansion formula which is r=0nncrxn-ryr. 

To determine the binomial coefficients, which is ncr= n!(n-r)! r!. The above binomial expansion formula can alternatively be expressed using this formula. 

(x+y)n has (n + 1) terms in its expansion. Let’s use the binomial expansion calculator is as follows: 

(x + y)n = xn + n xn – 1 y1 + n(n-1)2! xn-2 y2 +  n(n-1)(n-2)3!  xn – 3 y3 +… + nx1 yn – 1 + yn

Note: If we merely look at the coefficients, we can see that they are symmetric around the middle term. i.e., the first coefficient is identical to the last, the second coefficient is identical to the second from the last, and so on.

For Rational Powers

The expansion of (1+x)n is given by this binomial expansion formula, where the rational number is denoted by ‘n’. There are an unlimited number of terms in this expansion. 

(1+x)n= 1 + nx + n(n-1)2!x2 + n(n-1)(n-2)3!x3 + ….. 

Note: The value of |x| must be smaller than 1 to use this formula.

Important Terms of Binomial Expansion

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. The r-value, in this case, is one less than the number of binomial expansion terms. In addition, the coefficient is ncr , and the sum of the exponents of the variables x and y equals n.
• Middle Term: The total number of terms in the (x+y)n expansion equals n + 1. The binomial expression expansion’s middle term is determined by the value of n. The number of middle terms and the value of n is 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+1)/2 and (n+3)/2.
• Identifying a Particular Term: To find a phrase that contains xp, follow these two simple steps. Firstly, we must locate the general term in the expansion of (x + y)n, which is Tr+1=nCrxn−ryr. Second, we must compare this to xp to determine the r-value. The r-value can be used to locate a specific term in the binomial expansion. Let’s use the binomial theorem to find the fifth term in the expansion of (2x + 3)9.

The formula for calculating the nth term in a binomial expansion (x + y)n is Tr+1=nCr xn−ryr.

For 5th term r=4 

T5 = 9c4 (2x)9-4(3)4

• Term Independent of X: Finding a term independent of x follows the same processes as finding a specific term in the binomial expansion. First, we must locate the general term in the (x + y)n expansion.
• Numerically Greatest Term: The formula to discover the numerically greatest term in the expansion of (1+x)n  is (n+1)|x|1+|x|. There are two points to be remembered while using this binomial expansion formula to find the numerically greatest term. To begin, any binomial expansion must be converted into the form (1 + x)n. 

Conclusion 

Hopefully, you have gained a thorough understanding of the expansion of binomial expressions, their formulas, examples, and more. 

The Binomial Expression, as you can see, is a mathematical expression made up of two terms that include addition and subtraction operations. There’s no denying the fact that binomial expression is quite difficult to understand, especially when it comes to its formulas and equations. 

It’s incredibly important for you to go through the binomial expansion formulae and theorems to have a proper understanding of the topic to score well in your examination.