The Indian economy completely relies on the analysis of probability and statistics. These are widely used methods to obtain quick and efficient results, not only for our country but in all the countries. Therefore, to obtain the analysis, Binomial Theorem is heavily used.
The Binomial Theorem acts as a fundamental tool to gather specified results and helps perform higher-level mathematical calculations for any domain. The usage of the Binomial Theorem includes finding an equation’s roots in a higher power.
This article discusses the Maths important concept Binomial Theorem in detail while understanding all the other related concepts.
|Table of contents|
|Binomial Theorem – Definition|
|Usage of Binomial Theorem|
|Other useful expansions|
|Properties of Binomial Coefficients|
|Terms in Binomial Expansion|
|Middle Term in the x + yn . n|
|Determining a particular term|
|Numerically greatest term in the 1 + xn expansion|
|The ratio of consecutive terms or coefficients|
|Applications of Binomial Theorem|
|Book suggestions for studying Binomial Theorem|
Binomial Theorem – Definition
Binomial Theorem in CBSE Class 12 Mathematics states that for any provided positive integer n, the nth power of addition of two numbers x and y may be illustrated as the sum of the form of the term, n + 1. The Binomial Theorem acts as a simple formula that helps find any power of a specified binomial expression without needing to multiply the powers of the integers.
Usage of Binomial Theorem
In general, while dealing with higher-order integers, the expansion form would be very lengthy, and it would take forever to solve those equations. The calculation would become tedious and time-consuming. So, to make it easier for the Class 12 students to solve them, Binomial Theorem comes in handy.
For example, to simplify a binomial expression with higher powers, the Binomial Theorem is utilised, wherein the length of calculation becomes much shorter, saving time and effort.
In Binomial Theorem, one can expand an expression to any power. It is considered one of the most powerful tools while solving problems or equations in algebra, probability, statistics, etc.
- Binomial expression: A binomial expression is nothing but an algebraic expression that consists of two or more dissimilar terms. For example: a2+b2, a+b.
- Binomial Theorem: Let us consider n∈N, x, y∈R then,
x + yn = n r = 0 n Cr xn – r ·yr
n Cr = n !n – r ! r !
Given below are some of the important points to remember while expanding a binomial expansion:
- The variable ‘n’ is always the sum of the exponents of x and y.
- The total number of terms provided in the expansion of x + yn are n + 1.
- The binomial coefficients are n C0, n C1, n C2,…., n Cn. They can also be represented as Co, C1, C2,…., Cn.
- An important takeout while doing the binomial expansion is that the coefficients that are placed at an equal distance from the end as well as from the beginning are equal. For example, n C0 = n Cn, n C1 = n Cn – 1, n C2 = n Cn – 2 ,…. etc.
The below is Pascal’s Triangle which is used to find binomial coefficients.
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Other useful expansions
- x + yn – x – yn = 2 C1 xn – 1 y + C3 xn – 3 y3 + C5 xn – 5 y5 + …
- x + yn + x – yn = 2 C0 xn + C2 xn – 1 y2 + C4 xn – 4 y4 + …
- 1 + xn + 1 – xn = 2 [C0 + C2 x2 + C4 x4 + …]
- 1 + xn – 1 – xn = 2 [C1 x + C3 x3 + C5 x5 + …]
- 1 + xn = n r – 0 n Cr . xr = [C0 + C1 x + C2 x2 + … Cn xn]
- The number of terms contained in the x + an – x – an expansion is n2 if n is even or n + 12 if n is odd.
- The number of terms contained in the x + an + x – an expansion is n + 22 if n is even or n + 12 if n is odd.
Properties of Binomial Coefficients
These are denoted as the coefficients present in the binomial theorem. The following are the most important properties of binomial coefficients:
- C0 + C1 + C 2+ … + Cn = 2n
- C0 – C1 + C2 – C3 + … + – 1n . n Cn = 0
- C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n – 1
- n C1 + 2 . n C2 + 3 . n C3 + … + n . n Cn = n . 2n – 1
- C1 – 2 C2 + 3 C3 – 4 C4 + … + – 1n – 1 Cn = 0 for n > 1
- C02 + C12 + C22 …Cn2 = [ 2 n !n !2 ]
Terms in Binomial Expansion
While writing their Mathematics examination, the students of CBSE Class 12 are often asked to identify the general term or the middle term. The following are the various terms used in binomial expansion:
- Middle term
- General term
- Independent term
- Numerically greatest term
- Determining a particular term
- Ratio of consecutive terms or coefficients
Let us consider x + yn = n C0 xn + n C1 xn – 1 . y + n C2 xn – 2 . y2 + … + n Cn yn
General term = Tr + 1 = n Cr xn – r . yr
- General term in 1 + xn is n Cr xr
- In the x + yn binomial expansion, the rth term from the end is expressed as n – r + 2th
Middle Term in the x + yn . n expansion
- If n is an even number, then n2 + 1 term is considered as the middle term
- If n is an odd number, then n + 12 th and n + 32 th terms are defined as the middle terms.
Determining a particular term
- In the a xp + bxqn expansion, the coefficient of xm is also the coefficient of Tr + 1 where r = n p – m p + q .
- The other expansion is x + an,Tr + 1Tr = n – r + 1r .ax.
The term which is independent of, in the a xp + bxqn expansion is Tr + 1 = n Cr an – r br, where r = n p p + q (integer).
Numerically greatest term in the 1 + xn expansion
- If n + 1 x x + 1 = P + F, where P is denoted as a positive integer and 0 < F < 1, then P + 1th terms are considered as numerically greatest terms in the 1 + xn expansion.
- If n + 1 xx + 1 = P, is considered as a positive integer, then Pth term and P + 1th terms indicate they are the numerically greatest terms in the 1 + xn expansion.
Ratio of consecutive terms or coefficients
Coefficients of xr and xr + 1 are n Cr – 1 and n Cr respectively. Therefore:
n Crn Cr – 1 = n – r + 1r
Applications of Binomial Theorem
The usage of this theorem is humungous. Students of the CBSE Class 12 Mathematics should understand where this theorem is being applied. The following are some of the areas:
- The usage of the Binomial Theorem is mostly seen in statistical analysis and probability. Its usage is not just limited to these fields but is also used in many different sectors involving complex mathematical calculations.
- While solving higher degree mathematical calculations, the Binomial Theorem helps find the roots of higher power equations.
- Moreover, this theorem is also used in the analytical section of Physics and Science.
- It is used to rank candidates based on their performance in any firm.
- Helps identify calculations made during weather forecasting.
- Used in architecture while estimating costs during engineering projects.
Book suggestions for studying Binomial Theorem
The CBSE Maths students should be able to begin from the NCERT book. The demonstrations given in the book are very simple and lucid. Using this book, the Class 12 students can understand many things. They should start solving most of the practice and exercise problems given in the book. With this, the basic level of preparation would be accomplished.
After doing so, the students are advised to refer to a book called Cengage Mathematics Algebra. The explanation of the Binomial Theorem has been given in a very clear way such that it would be easy for the students to grab the concept, along with several model questions.
Apart from that, the CBSE students can also use the book called Arihant Algebra by SK Goyal or RD Sharma. However, the choice of reference differs from one student to the other, depending on their intellectual capacities. The students should find the best-suited book based on their convenience.
To conclude, the usage of the Binomial Theorem in the world is tremendous. It is the same in the case of the Class 12 board exam as well. The students should spend ample time focusing on this topic to obtain maximum marks.
With the help of this article, the CBSE Maths students were able to learn how to use the Binomial Theorem and understand its applications in various domains. Moreover, a topic called Binomial Expansion is also being discussed. In addition to this, the students were taught the properties of this theorem.
The next topic we discussed is the various types of terms used in the Binomial Theorem, where students were educated on all the different terms in detail. Finally, we have also seen what sets of books one needs to use to comprehend this theorem.
1. What are some of the important topics in the Binomial Theorem?
The below mentioned are some of the fundamental topics of the Binomial Theorem:
- Pascal’s Triangle
- General term
- Middle term
- Binomial Theorem for positive integral index
- Properties and application of Binomial Theorem
2. What are the real-time applications of this theorem?
Below mentioned are some of the places where the Binomial Theorem is vastly used:
- Statistics analysis
- Weather forecasting
- Engineering purposes such as Architecture
3. How to study the Binomial Theorem?
The students should be able to comprehend the Binomial Theorem and then try finding the general term and greatest term of any binomial equation. Once the students have constructed a strong foundation on the theorem, they can then go on and learn about the application section.
While preparing for the CBSE Class 12 boards, the students should solve as many exercise problems and mock tests as possible. This would give them confidence and an overall idea of the theorem. One thing every student needs to remember is that, if they have made a strong foundation for this topic, then it would be a piece of cake to solve this topic in the examination.
4. Mention the important books to refer to while learning Binomial Theorem.
The students need to follow some books to achieve higher scores in the examination, especially in the Binomial Theorem topic. They are:
- NCERT (for basic understanding)
- Arihant Algebra by RD Sharma or SK Goyal
- Cengage Mathematics Algebra
5. What are the three types of terms in the binomial expansion?
The three types of terms in binomial expansion are:
- Monomial term
- Binomial term
- Trinomial term