## About Binomial Theorem Class 11 Chapter 8

The NCERT Solutions for Binomial Theorem Class 11 Chapter 8 can be downloaded at Vedantu without any hassle. Practicing these Binomial Theorem Class 11 solutions can help the students clear their doubts as well as solve the problems faster. Students can learn new tricks to answer a particular question in different ways giving them an edge with exam preparation.

The concepts of Binomial Theorem Class 11 covered in Chapter 8 of the Maths textbook include the study of essential topics, such as Positive Integral Indices, Pascal’s Triangle, Binomial theorem for any positive integer, and some special cases. Students can score high marks in the exams with ease by practicing the Binomial Theorem Class 11 solutions for all the questions present in the textbook. It gets pretty apparent to know the logic set behind each answer and develop a far better comprehension of the concepts.

### NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8

All the Binomial Theorem Class 11 solutions Maths Chapter 8 is given here. This solution also contains questions, answers, images, and explanations for the whole chapter 8 titled with the Binomial Theorem taught in Class 11. If you are a student of Class 11th who is referring to the NCERT book to study Mathematics, then you will come across chapter 8 Binomial Theorem after you have completed studying the lesson, you also must be looking for the answers to the questions. You will get here all the Binomial Theorem Class 11 solutions for Mathematics Chapter 8 all in a single place.

### Definition of Binomial Theorem

Sometimes, when the power increases, the expansion becomes lengthy and tedious to calculate. A binomial expression that has been raised to a really large power is often easily calculated with the assistance of the theorem.

### Binomial Expansion

Important points to remember

The total number of the terms in the expansion for (x+y)n is (n+1)

The sum of the exponents of x and y is always n.

nC

_{0}, nC_{1}, nC_{2}, … .., nC_{n}are called binomial coefficients and also are represented by C_{0}, C_{1}, C_{2}, ….., C_{n}The binomial coefficients which are equidistant from the beginning and from the ending are also equal i.e. nC

_{0}= nC_{n}, nC_{1}= nC_{n-1}, nC_{2}= nC_{n-2},….. etc.

### Terms in the Binomial Expansion

In the binomial expansion, it is normally asked to find the middle term or the general term. Different terms in the binomial expansion that are covered are included here:

General Term

Middle Term

Independent Term

Determining a Particular Term

Numerically greatest term

The ratio of Consecutive Terms/Coefficients

### What is a Binomial Expression?

A binomial is also a mathematical expression that includes two terms. Further, these two terms must be separated from each other by either addition or subtraction. To add the binomials, one should combine equal terms to get an answer. To multiply the binomials, the distributive property should be used.

### Topics and SubTopics of Binomial Theorem

### Binomial Theorem Formula

The Binomial Theorem is a quick method for expanding or multiplying out of a binomial expression. The expression also has been raised to some bigger power. As we know the multiplication of such expressions is always difficult with large powers. But the Binomial expansions and their formulas help us a lot in this regard. In this article, we will discuss the Binomial theorem and its Formula.

( a + b )^{n} = k =0n(k^{n}) a^{k} b^{n-k}

The upper index n is known as the exponent for the expansion; the lower index k points out which term, starting with k equals 0. For example, when n equals 5, each of the terms in the expansion for (a + b)^{5} will look like: a^{5} − ^{k}b^{k}.

### Properties of the Binomial Expansion (x + y)^{n}

There are a total of n+1 terms.

The first term is known to be (x)^{n}

Progressing from the first term to the last term, the exponent for x decreases by 1 from term to term. Whereas the exponent of y increases by 1 term. Also, the sum for both the exponents in each term will be n.

If the coefficient of each of the terms is multiplied by the exponent of x in that term, and the product is divided by the number of that term, we can easily get the coefficient for the next term.

### Solved Questions

Expand the following terms using binomial theorem.

(1-2x)

^{5}

Using the binomial theorem, we will expand the above given expression as: (1-2x)^{5} = 5C0(1)^{5} - 5C1(1)^{4} (2x) + 5C2(1)^{3} (2x)^{2 }- 5C3(1)^{2} (2x)^{3} + 5C4(1)^{1} (2x)^{4} - 5C5 (2x)^{5}

= 1 - 5(2x) + 10 (4x)^{2} -10 (8x^{3}) +5 (16x^{4}) - (32 x^{5})

= 1 - 10x + 40x^{2} - 80x^{3} + 80 x^{4} -32 x^{5}

This is the final expanded form of (1-2x)^{5}, found using binomial theorem.

(2x-3)

^{6}

Using binomial theorem, the following term can be expanded as follows:

(2x-3)^{6} = 6C0 (2x)^{6} - 6C1 (2x)^{5}(3)1 + 6C2 (2x)^{4}(3)^{2} - 6C3 (2x)^{3}(3)^{3} + 6C4 (2x)^{2}(3)^{4} - 6C5 (2x)^{1}(3)^{5} + 6C6(3)^{6}

= 64 x^{6} - 6(32 x^{5}) (3) + 15 (16 x^{4})(9) - 20(8 x^{3}) (27) + 15(4x^{2})(81) - 6(2x)(243) + 729

= 64 x^{6} - 576 x^{5} + 2160 x^{4} - 4320 x^{3} + 4860 x^{2} - 2916 x + 729.

This is the final expanded form of (2x-3)^{6}, found using the binomial theorem.

Solve (101)

^{4}using the binomial theorem

Any number like 101 can be expressed as the sum or difference of two numbers whose powers are easy to calculate. Using Binomial Theorem the value can be evaluated as :

101 can be written as 100+1

So 101^{4} = (100 +1)^{4}

(100 +1)^{4} = 4C0(100)^{4} + 4C1(100)^{3} (1) + 4C2(100)^{2} (1)^{2} + 4C3(100)^{1} (1)^{3} + 4C4 (1)^{4}

= (100)^{4}+ 4 (100)^{3} + 6 (100)^{2} + 4(100) +1

= 104060401

### Did You know?

Generally, Sir Issac Newton is credited for the work of the binomial theorem.

The binomial theorem is highly used in statistical and probability analysis.

### Study Tips

Math is often considered difficult to study as compared to other subjects.

However, it should not be under-determined that after consecutive efforts and solving enough practice problems, one can score good marks in maths.

Binomial Theorem is another topic in mathematics for which minimal memorization is required

First, understand the concept and then remember the important expansion formulas.

Once the concept is being understood, then go to solve practice problems as much as possible.

There are many online websites that give solutions and questions for NCERT or any of these State Board exams.

Practicing more questions of Binomial Theorem will give more confidence in solving questions from Binomial topic

This was all about binomial theorem formulas, properties and FAQs. For more such information, access free resources available on the Vedantu website useful for the state board, CBSE, ICSE, and competitive examinations. All NCERT Solutions for all subjects are available on the Vedantu website.

## FAQs on Binomial Theorem Class 11

**1. What is a Binomial Theorem?**

The binomial theorem is valid normally for any of the elements x and y of a satisfying xy equal yx. The theorem is true even more: alternativity suffices in place of associativity. The binomial theorem can also be said in a way that the polynomial sequence here given as {1, x, x^{2}, x^{3}, ...} is of a binomial type.

**2. What is K in the Binomial Theorem?**

The upper index n is known as the exponent for the expansion; the lower index k points out which term, starting with k equals 0. For example, when n equals 5, each of the terms in the expansion for (a + b)^{5} will look like: a^{5} − kbk.

**3. Where is the Binomial Theorem Used?**

The theorem and its generalizations also can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas in mathematics. The binomial theorem also helps explore probability in an organized way: someone says that he/she will flip a coin 5 times.

**4. What is a First Degree Binomial?**

Examples: 5x^{2}-2x+1 The highest exponent here is 2 so this is a 2^{nd} degree trinomial. 3x^{4}+4x^{2} The highest exponent is 4 here so this is a 4^{th} degree binomial. 8x-1 It appears that there is no exponent, but the x has an understood exponent of 1; So, this is known to be a 1st degree binomial.