Binomial Theorem

Using the direct multiplication we may expand , but as the power increases the expansion becomes lengthy and tedious to calculate.

A binomial expression(an expression containing two unlike terms, eg. ()) that has been raised to a very large power can be easily calculated with the help of the Binomial Theorem.Let us try to understand Binomal Theorem in detail in this article.

Table of Contents:

• Binomial Theorem
• General Term
• Some Important Points Related to Binomial Theorem
• Concept Video
• Pascal’s Triangle
• Concept Video
• Some Important Expansions
• Middle Term
• Concept Video
• Practice Problems on Binomial Theorem
• FAQs on Binomial Theorem

Binomial Theorem

Binomial Theorem is the method of expanding an expression that has been raised to any finite power. The binomial theorem has the expansion as follows:

,

Here,  and

  ( + 1) terms

We can observe that,

General Term

  term in the expansion of is known as a general term. Given by,

Some Important Points Related to Binomial Theorem

• Number of terms in the expansion of
• Sum of powers in each term of the expansion is n.
•  are known as binomial coefficients.
• The binomial coefficients of the terms equidistant from the beginning and the end are equal  i.e.,
• Sum of all binomial coefficients  in the expansion of  is .

Proof :

 Putting  in above equation, we get

• Sum of even term binomial coefficients = Sum of odd term binomial coefficients

Proof : Put  in the expansion formula i.e.

 (sum of even or odd term binomial coefficients)

 Sum of even or odd term binomial coefficients 

• Coefficients and binomial coefficients are different.

            Example:

            Here, sum of coefficients

            where as, sum of binomial coefficients 

•   
• If    
• The  term from the end in the expansion of is  term from the beginning.

Concept Video:

Properties of Binomial Coefficients | MATHS | JEE | Concept of the Day | GB Sir

Pascal’s Triangle

Pascal’s Triangle is basically an arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle. Firstly, 1 is placed at the top, and then we start putting the numbers below in a triangular pattern. The numbers which we get in each step are the addition of the above two numbers.

Concept Video:

Pascal's Triangle | MATHS | JEE | Concept of the Day | GB Sir

For example, the pascal’s triangle for the expansion of  is shown below:

Some Standard Expansions

• ,

• ,

,

• ,

Example : Expand

Solution :

We know the binomial expansion as follows:

Here,

 Now, }}

  is the required expansion.

Middle Term

In the expansion of  middle term can be found for two cases

Case ' is even

Total Number of terms  (odd)

Hence, middle term is

Case II :  is odd

Number of terms (even)

Hence, two middle terms are

Concept Video:

General and Middle Terms of Binomial Expansion | MATHS | JEE | Concept of the Day | GB Sir

Numerically greatest term in the expansion of

• If  , where m is a positive integer then  will be the Numerically greatest terms (Both terms are equal)

• If , where  is a positive integer and then   is numerically the greatest term in the expansion of .

Practice Problems on Binomial Theorem

Example : If the third term in the binomial expansion of  is  then find a possible value of x.

(a)            (b)                (c)               (d)  

Answer:

Example :

Find the term independent of  in the expansion of

Answer:

Example : If  where  and , then which of the following is/are true?

(a) is even   (b)        (c) is odd         (d)

Answer:

Given, 

Let

By adding equation (i) and (ii), we get,

Also, given that,

So,

Using  

In the above expression   . So, we have,

As we can see that in the above expression,  side is an even natural number.

Hence,= should also be an even natural number.

As =.So, =

Also, =, i.e., odd       (So that L.H.S is even)

=

=and=

As we can see that, =

or =

Hence, options (c) and (d) are the correct answers.

Example : If , then find the value of .

a)                 b)              c)                d)  

Answer:

Given,   

Let

Hence, option (b) is the correct answer.

Example : Find the value of , where  is an even positive integer.

Answer:

Let

FAQs on Binomial Theorem

Question 1.What is the method to prove the Binomial Theorem formula?

Answer: We can apply Principle of Mathematical Induction to prove the Binomial Theorem formula.

Question 2.What is the use of Binomial expansion formula?

Answer: Binomial expansion is used to calculate the high powers of a Binomial expression without actually using the method of direct multiplication.

Question 3.What is the total number of terms in .

Answer: The total number of terms is given by  .

Question 4.Is Binomial theorem for fractional index also defined?

Answer: Yes, there is a separate binomial theorem for fractional index (power)