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Binomial Theorem

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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>3</mn><mi>x</mi></mfrac></mrow></mfenced><mn>4</mn></msup></math>

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

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced open="[" close="]"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>&#xA0;</mo><mo>=</mo><msup><mfenced open="[" close="]"><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mo>&#xA0;</mo><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mspace linebreak="newline"/><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo>,</mo><mo>&#xA0;</mo><msub><mi>T</mi><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mi>n</mi><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mi>n</mi><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></math>

Case II :  is odd

Number of terms (even)

Hence, two middle terms are

 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mo>&#xA0;</mo><mo>&amp;</mo><msup><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>t</mi><mi>h</mi></mrow></msup><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo>,</mo><mo>&#xA0;</mo><msub><mi>T</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><mspace linebreak="newline"/><mo>&amp;</mo><mo>&#xA0;</mo><msub><mi>T</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac></msub><mo>=</mo><mmultiscripts><mi>C</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><none/><mprescripts/><none/><mi>n</mi></mmultiscripts><msup><mi>x</mi><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup><msup><mi>y</mi><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></msup></math>

Concept Video:

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

Numerically greatest term in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math>

  • If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced open="|" close="|"><mi>x</mi></mfenced></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>m</mi></math> , where m is a positive integer then  will be the Numerically greatest terms (Both terms are equal)

 

  • If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced open="|" close="|"><mi>x</mi></mfenced></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>m</mi><mo>+</mo><mi>f</mi><mo>&#xA0;</mo></math>, 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:

 

 

Step 1 :

 

We know the general term in the binomial expansion of ( is

So, third term,

Put

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><none/><mprescripts/><none/><mn>5</mn></mmultiscripts><msup><mfenced><mn>1</mn></mfenced><mn>3</mn></msup><msup><mfenced><msup><mi>x</mi><mrow><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mn>2</mn></msup><mo>=</mo><mn>2560</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mn>10</mn><mfenced><msup><mi>x</mi><mrow><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mo>=</mo><mn>2560</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mfenced><msup><mi>x</mi><mrow><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mo>=</mo><mn>256</mn><mspace linebreak="newline"/><mi>N</mi><mi>o</mi><mi>w</mi><mo>,</mo><mo>&#xA0;</mo><mi>l</mi><mi>e</mi><mi>t</mi><mo>&#xA0;</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>t</mi><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><msup><mn>2</mn><mi>t</mi></msup><mspace linebreak="newline"/><mo>&#x21D2;</mo><mfenced><msup><mfenced><msup><mn>2</mn><mi>t</mi></msup></mfenced><mrow><mn>2</mn><mi>t</mi></mrow></msup></mfenced><mo>=</mo><mn>256</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><msup><mfenced><mn>2</mn></mfenced><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mo>=</mo><mn>256</mn></math>

Step 2 :

 

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><none/><mprescripts/><none/><mn>5</mn></mmultiscripts><msup><mfenced><mn>1</mn></mfenced><mn>3</mn></msup><msup><mfenced><msup><mi>x</mi><mrow><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mn>2</mn></msup><mo>=</mo><mn>2560</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mn>10</mn><mfenced><msup><mi>x</mi><mrow><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mo>=</mo><mn>2560</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mfenced><msup><mi>x</mi><mrow><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced></mrow></msup></mfenced><mo>=</mo><mn>256</mn><mspace linebreak="newline"/><mi>N</mi><mi>o</mi><mi>w</mi><mo>,</mo><mo>&#xA0;</mo><mi>l</mi><mi>e</mi><mi>t</mi><mo>&#xA0;</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>t</mi><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><msup><mn>2</mn><mi>t</mi></msup><mspace linebreak="newline"/><mo>&#x21D2;</mo><mfenced><msup><mfenced><msup><mn>2</mn><mi>t</mi></msup></mfenced><mrow><mn>2</mn><mi>t</mi></mrow></msup></mfenced><mo>=</mo><mn>256</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><msup><mfenced><mn>2</mn></mfenced><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mo>=</mo><mn>256</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x21D2;</mo><msup><mfenced><mn>2</mn></mfenced><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mo>=</mo><msup><mn>2</mn><mn>8</mn></msup><mspace linebreak="newline"/><mo>&#x21D2;</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>t</mi><mo>=</mo><mo>&#xB1;</mo><mn>2</mn><mspace linebreak="newline"/><mo>&#x2235;</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>t</mi><mspace linebreak="newline"/><mo>&#x21D2;</mo><msub><mi>log</mi><mn>2</mn></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>&#xB1;</mo><mn>2</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><msup><mn>2</mn><mrow><mo>&#xB1;</mo><mn>2</mn></mrow></msup><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>x</mi><mo>=</mo><mn>4</mn><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mspace linebreak="newline"/></math>

Hence, the correct option is (b).

Example :

Find the term independent of  in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mo>&#xA0;</mo></mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mo>&#xA0;</mo></mrow><mfenced><mrow><mi>x</mi><mo>-</mo><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></msup></mrow></mfenced></mfrac></mrow></mfenced><mn>10</mn></msup><mo>,</mo><mo>&#xA0;</mo><mi>x</mi><mo>&#xA0;</mo><mo>&#x2260;</mo><mo>&#xA0;</mo><mn>1</mn></math>

Answer:

 

Step 1 :

 

Simplifying the given expression:

 

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mo>&#xA0;</mo></mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mo>&#xA0;</mo></mrow><mfenced><mrow><mi>x</mi><mo>-</mo><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></msup></mrow></mfenced></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><msup><mi>x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mfenced><mrow><msup><mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfenced><mn>3</mn></msup><mo>+</mo><msup><mn>1</mn><mn>3</mn></msup></mrow></mfenced></mfrac><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfrac><mrow><mo>(</mo><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfrac><mfenced><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mfrac></mrow></mfenced><mn>10</mn></msup><mspace linebreak="newline"/><mo>=</mo><msup><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>-</mo><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>10</mn></msup></math>

 

Step 2 :