General Term in Binomial Expansion

A binomial expression is a mathematical tool used in finding and expressing the expanded indication of any algebraic expression raised to a power and is in the form of (a+b)n. However, you might be thinking about why we need a binomial theorem to expand an algebraic expression. You can just do that by applying simple formulas. 

The usage of methods of binomial expansions can be understood by expressions such as (a+b)190. In this particular expression, you would find it very difficult to obtain the expanded form. This is where binomial expansion comes in, and it saves your time and energy by framing the final equation using a simple formula. 

Binomial Expansion Formula. 

Binomial in algebra is the name given to a pair of terms separated by mathematical operations such as addition or subtraction. A binomial theorem is employed to derive the expanded form of any expression in (a+b)n form. To derive the final formula, you must understand the binomial theorem. 

Binomial Theorem. 

The binomial theorem states that “According to the binomial theorem, it is possible to expand any power of binomial (x + y) into a sum of its various terms.” Based on this theorem, mathematicians have derived the binomial expansion formula. 

Formula For Binomial Expansion.

You can write the binomial expansion formula of an expression (p+q)^n   (For now, n >=0 )as 

(p+q)^n = nC0p^n + nC1p^n-1q^1 + nC2p^n-2q^2 +………. + nCr p^n-rq^r +…… + nCnq^n

General Term in Binomial Expansion. 

A binomial expression is a set of all the numbers obtained when a number raised to a positive power is expanded. However, certain binomial expressions are too long that their expansion would include a wide array of numbers. Therefore, you need to have a way to get the exact term in a binomial expression that you seek. 

As we have discussed that the general formula for a binomial expression for an integer n is 

(p+q)^n = nC0p^n + nC1p^n-1q^1 + nC2p^n-2q^2 +………. + nCr p^n-rq^r +…… + nCnq^n

Therefore, we can use the following formula to get the general term in binomial expansion.

For the expansion of (a+b)n the general term will be ( Tr+1).

so ,  T^r+1 =  nCra^n-rb^r  for r = 0, 1, 2……, n

One common mistake that most people commit in applying this expression is that they ignore that the first term of any binomial expression will take r = 0. So if you are putting 1 in the general term to find the first term, you will come up with a wrong number. 

Coefficients In Binomial Expressions. 

Every term of a binomial expansion contains a coefficient attached to the variable. To obtain the coefficient of any binomial expression, you need to use Pascal’s triangle or combination method. 

Pascal’s Triangle. 

All the binomial coefficients can be arranged in the form of a triangle that any given number in the triangle is the sum of two numbers that exist above it. Renowned scientist Blaise Pascal proposed this method of putting the coefficients in a triangle. Another feature to note is that the borders of any Pascal triangle will only contain 1. 

nCr = n! r!( n-r)!

Moreover, every coefficient obtained using this method follows these properties. 

1. nC0 = nCn = 1
2. nC1 = nCn-1 = n
3. nCr =  nr n-1Cr-1 

All of these aforementioned properties can be easily obtained from the binomial expansion of an algebraic expression (x+y)^n.  

Properties Of Binomial Expansion

All Binomial expressions have some properties which make it easier for mathematicians to identify them. Some of these properties are – 

• The total number of terms in the binomial expansion of (a+b)^n  will be n+1

• The first and last term in the binomial expansion is an and bn, respectively

• In every binomial expression, the power of the first term, i.e. a, decreases from n to 0, while the power of the second term, i.e. b, increases from 0 to n

• All the coefficients that are used in a binomial expression can be arranged in a uniform way using Pascal’s triangle

Binomial Expansion Calculator

To obtain binomial expressions of algebraic expressions using the binomial theorem, you can use a binomial expansion calculator. In this calculator, all you need to do is add the term in the search box with its coefficient in the subsequent box. Entering the value, you will obtain the expanded form of that algebraic expression. 

Conclusion

A binomial expression is one that has two terms joined by a mathematical operation, such as addition or subtractions. The binomial theorem is used to derive the binomial expansion formula, which is used to find the expanded form of any binomial expression with positive or negative power. This formula is written as (p+q)^n = nC0p^n + nC1p^n-1q^1 + nC2p^n-2q^2 +………. + nCr p^n-rq^r +…… + nCnq^n

Most binomial expansions are very vast, and to find the exact term in the sequence, one uses the formula for the general term in Binomial expansion. To understand this formula more vividly, one needs to have a clear understanding of coefficients and different types of terms such as constant terms etc.