Binomial Theorem Solutions - Formula, Worksheet, Calculator and Examples.

Binomial Theorem Questions - Page 4
Binomial Theorem Video Lecture - Page 5

However, for higher powers like 98^{2},101^{6} , etc., the calculations become difficult by using repeated multiplication. This difficulty was overcome by a theorem known as binomial theorem. It gives an easier way to expand , (a+b)^{n} where n is an integer o r a rational number.

Binomial Theorem for Positive Integral Indices

Let us have a look at the following identities done earlier

(a+b)^{0}=\: \: 1 a + b ≠ 0

(a+b)^{1}=\: \:a+b

(a+b)^{2}=\: \:a^{2}+2ab+b^{2}

(a+b)^{3}=\: \:a^{3}+3a^{2}b+3ab^{2}+b^{3}

(a+b)^{4}=\: \:(a+b)^{3}(a+b)\: =\: a^{4}4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}

In these expansions, we observe that

  • The total number of terms in the expansion is one more than the index. For example, in the expansion of (a+b)^{2} , number of terms is 3 whereas the index of (a+b)^{2} , is 2
  • Powers of the first quantity ‘ a ’ go on decreasing by 1 whereas the powers of the second quantity ‘ b ’ increase by 1, in the successive terms.
  • In each term of the expansion, the sum of the indices of a and b is the same and is equal to the index of a + b
    We now arrange the coefficients in these expansions as follows (Fig8.1):

Do we observe any pattern in this table that will help us to write the next row? Yes we do. It can be seen that the addition of 1’s in the row for index 1 gives rise to 2 in the row for index 2. The addition of 1, 2 and 2, 1 in the row for index 2, gives rise to 3 and 3 in the row for index 3 and so on. Also, 1 is present at the beginning and at the end of each row. This can be continued till any index of our interest.

We can extend the pattern given in Fig 8.2 by writing a few more rows.

 

Latest Jobs

  1. Type
    Job Notification
    Job
    KVIC Recruitment of 108 Group-B and Group-C categories - Last Date 19-01-2020
    Location
    MumbaiMaharashtra, India
    Date Posted
    7 Jan 2020
  2. Type
    Job Notification
    Job
    SAIL Durgapur Steel Plant Recruitment of Manager, Technician, Medical Service Provider - 27-01-2020
    Location
    DurgapurWest Bengal, India
    Date Posted
    7 Jan 2020
  3. Type
    Job Notification
    Job
    Indian Coast Guard Recruitment 260 Navik - Last Date 02-02-2020
    Location
    Anywhere
    Date Posted
    7 Jan 2020
  4. Type
    Job Notification
    Job
    IOCL hiring 312 Apprentices - Last Date 22-01-2020
    Location
    Anywhere
    Date Posted
    7 Jan 2020
  5. Type
    Job Notification
    Job
    RBI Recruitment NEWS for 926 Assistants - Last Date 16-01-2020
    Location
    Anywhere
    Date Posted
    7 Jan 2020
  6. Type
    Job Notification
    Job
    Janki Devi Memorial College Recruitment 71 Assistant Professor - Last Date 04-01-2020
    Location
    DelhiDelhi, India
    Date Posted
    2 Jan 2020
  7. Type
    Job Notification
    Job
    NPCIL Recruitment of Trainee, Electrician, Fitter, Machinist, Welder, Driver - Last Date 21-01-2020
    Location
    RawatbhataRajasthan, India
    Date Posted
    2 Jan 2020
  8. Type
    Job Notification
    Job
    AIIMS Nagpur 104 Principal, Lecturer, Staff Nurse. Last Date 14-01-2020
    Location
    NagpurMaharashtra, India
    Date Posted
    2 Jan 2020
  9. Type
    Job Notification
    Job
    Delhi Police Recruitment of 649 Head Constable - Last Date 27-01-2020
    Location
    DelhiDelhi, India
    Date Posted
    2 Jan 2020
  10. Type
    Job Notification
    Job
    IBPS Recruitment Specialist Officers 1163 Vacancies - Last Date 26-11-2019
    Location
    Anywhere
    Date Posted
    20 Nov 2019

Jobs by Category

Syllabus and Previous Papers

Leave a Reply

Your email address will not be published. Required fields are marked *