# Higher Algebra Hall & Knight Binomial Theorem Positive Integral Index (Chapter 13) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 13 ‘Binomial Theorem, Positive Integral Index’ will help you understand all the concepts of the chapter for JEE Mains and JEE Advanced preparation. These solutions discuss different topics like a binomial theorem, binomial expansion, various terms of binomial expansion like the middle term, the general term, independent term or the nth term. The chapter renders you with different formulas that help you in finding the terms quickly by not using the long multiplication methods.

Higher Algebra by Hall and Knight Chapter 13 Binomial Theorem, Positive Integral Index contains 2 exercises in which you need to solve questions that range from basic to advanced level. There are 58 questions in this chapter based on finding the general term of a binomial expansion or the nth term like finding the 4th or 6th term in a binomial expression along with finding the greatest coefficient in a binomial expansion. All these questions will help you prepare for IIT JEE and NEET competitive exams.

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## Important Topics for Hall and Knight Higher Algebra Solutions Chapter 13: Binomial Theorem Positive Integral Index

This chapter is about binomial theorems. Before understanding the theorems, it is essential to understand the meaning of binomial. The word ‘Binomial’ is a special case of the word – ‘Polynomial’ that is an algebraic expression containing two or more algebraic terms. Thus, the binomial also contains exactly two different terms.

**Binomial Theorem**

Binomial Theorem is mainly the method that helps in expanding an expression that is raised to any finite power. The binomial theorem has its application in probability, algebra, and other fields of mathematics. This helps in finding the power of a binomial without using the long methods of multiplication.

**Binomial Expression**

Binomial means a mathematical expression that is divided into two terms through addition or subtraction. In order to add the binomials, you will have to combine the like terms to find the right answer. In order to multiply the binomials, you will have to apply distributive property as you will never find an answer through multiplication.

Binomial Theorem for positive integral index states that the total number of terms in the expansion of the theorem is always one more than the index. Let us understand this with an example:

In the expansion of (a + b )^{n} , the number of terms is n+1, whereas the index of (a+b)^{n} is n, where n is any positive integer.

**Terms of Binomial Expansion **

There are different terms of binomial expansion which are listed below:

**General Term **

With the formula of the binomial theorem, you know that there are (n+1) terms in the expansion of (a+b)^{n}. For instance, T_{1}, T_{2}, …… T_{n+1}, are the first second and (n +1)th terms in the expansion of (a+b)^{n}.

The general term in the expansion of (a + b)n, is given by:

T_{r+1} = a_{r+1} = nC_{r} a^{n-r} b^{r}

**Middle term in the expansion of (x+y) ^{n.n}**

When you will try to expand (a + b)^{n}, and ‘n’ is an even number, then (n + 1) will be an odd number. This means that in the expansion, you will get an odd number of all the terms. In this case, the middle term will be ( n/2 + 1)th term.

For example, if you will expand, ( x + y )^{3}, then the middle terms will be, ( 3+1/ 2) = 2nd term and ( 3+3/2) = 3rd term.

**Independent term **

Independent term is a term in a binomial expansion in which the power of x is zero. In order to find the independent term of x, you need to set the power of x to zero. In the chapter, you will understand this concept with examples.

Other terms of binomial expansion covered in the chapter are:

- Determining a particular term
- Numerically greatest term
- Ratios of consecutive terms/coefficients

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 13: Binomial Theorem Positive Integral Index

- Higher Algebra by Hall & Knight Book Chapter 13 contains two exercises with 58 questions.
- In the first exercise, you need to expand the binomials, simplifying the binomials by finding the right number of the term as asked in the question, finding the value of middle, general, independent or a number term like 4th, or 5th term.
- In the second exercise, you need to identify the greatest term, find out the greatest term in an expansion, finding the rth term from the end, showing the middle term to be A or B as given in the question and much more.
- The questions in the book might make maths complex for you but with proper study, you will be able to solve the advanced level questions.

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