# Binomial Theorem, Objectives, the Binomial Theorem for a Natural Exponent

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• The binomial theorem, was known to Indian and Greek mathematicians in the rd century B.C. for some cases.

• The credit for the result for natural exponents goes to the Arab poet and mathematician Omar Khayyam (A.D. ). Further generalisation to rational exponents was done by the British mathematician Newton (A.D. ).

## Objectives

After studying this lesson, you will be able to:

## The Binomial Theorem for a Natural Exponent

You must have multiplied a binomial by itself, or by another binomial. Let us use this knowledge to do some expansions. Consider the binomial. Now,

and so on.

In each of the equations above, the right hand side is called the binomial expansion of the left hand side.

Note that in each of the above expansions, we have written the power of a binomial in the expanded form in such a way that the terms are in descending powers of the first term of the binomial (which is x in the above examples). If you look closely at these expansions, you would also observe the following:

. The number of terms in the expansion is one more than the exponent of the binomial. For example, in the expansion of , the number of terms is .

. The exponent of in the first term is the same as the exponent of the binomial, and the exponent decreases by in each successive term of the expansion.

. The exponent of in the first term is zero (as). The exponent of in the second term is , and it increases by in each successive term till it becomes the exponent of the binomial in the last term of the expansion.

. The sum of the exponents of and in each term is equal to the exponent of the binomial. For example, in the expansion of , the sum of the exponents of and in each term is .

If we use the combinatorial co-efficient, we can write the expansion as

, and so on.

More generally, we can write the binomial expansion of , where is a positive integer, as given in the following theorem. This statement is called the binomial theorem for a natural (or positive integral) exponent.

Theorem:

where and .

Proof:

Let us try to prove this theorem, using the principle of mathematical induction.

Let statement be denoted by , i.e.,

Let us examine whether is true or not.

From, we have

i.e., Thus, holds.

Now, let us assume that is true, i.e.,

Assuming that is true, if we prove that is true, then holds, for all . Now,

i.e.

From Lesson, you know that and

Also,

Therefore,

……………………….

.……………………… and so on

Using and , we can write as

which shows that is true.

Thus, we have shown that is true, and if is true, then is also true.

Therefore, by the principle of mathematical induction, holds for any value of . So, we have proved the binomial theorem for any natural exponent.

This result is supported to have been proved first by the famous Arab poet Omar Khayyam, though no one has been able to trace his proof so far.

Example:

Write the binomial expansion of .

Solution:

Here the first term in the binomial is and the second term is . Using the binomial theorem, we have

Thus,