Education

Binomial Theorem

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What is Binomial?

A Binomial is a Polynomial with two expressions. For example, 2x3 + 3 is a binomial.

The Binomial Theorem describes what happens when binomial is multiplied as many number of times by itself. But there has to be a specific pattern for it, and to understand about it, you need to know what exactly an exponent means.

Exponent

Exponent shows how many times you want to multiply something.

                          Example, 52 = 5 x 5 = 25

Here, 2 is known as exponent of 5(known as base).

The other name for exponent is index, or power.

Please note that an exponent of value 0 is not used and considered as 1 on a whole.

                                                  Example, 50 = 1

Exponents of (a + b)

Now we can move to binomial. Let’s start with simple binomial a+b, that can be any binomial.

To begin with an exponent 0, and move upward.

Exponent of 0

You get the resultant value as 1 for any value of the binomial when the exponent is 0.

(a + b)0 = 1

Exponent of 1

You conclude with the original value itself when the exponent is 1.

(a + b)1 = a + b

Exponent of 2

An exponent of 2 describes to multiply by itself.

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

Exponent of 3

You need to multiply one more time with the original value when the exponent is 3.

(a + b)3 = (a + b) (a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3

Now you must have got enough idea to frame a pattern.

According to the last result, we got:

a3 + 3a2b + 3ab2 + b3

Note the exponents of a. It starts with 3 and goes down as 3, 2, 1, 0.

Similarly, the exponents of b goes up as 0, 1, 2, 3.

If you number the terms as 0 to n, we get the following.

K = 0 K = 1 K = 2 K = 3
a3 a2 a 1
1 b b2 b3

To bring this table in a single frame, we get

an-kbk

For example, let’s take when exponent, n = 3

K = 0 K = 1 K = 2 K = 3
an-kbk an-kbk an-kbk an-kbk
a3-0b0 a3-1b1 a3-2b2 a3-3b3
= a3 =a2b =ab2 =b3

Coefficients

Let’s consider all the results we obtained from (a + b)0 to (a + b)3

1

a + b

a2 + 2ab + b2

a3 + 3a2b + 3ab2 + b3

Now, look only the coefficients with a 1 wherein the coefficient does not exist.

1

1 + 1

1 + 2 + 1

1 + 3 + 3 + 1

This is actually a Pascal’s Triangle, wherein each number is the sum of the above two numbers except at edges which are 1.

Formula

The final step is to place all the terms in one single formula. A formula that can be applied to as many number of terms required. Therefore, the handy Sigma Notation can be used in these cases.

The formula is as shown below.

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Where, k can be any value from 0 to n and known as the Binomial Theorem.

Let’s solve an example.

Try it for n = 3:

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It is much simple to remember the pattern.

  • The first term’s exponents begins at n and goes down.
  • The second term’s exponents begins at 0 and goes up.

  BYJU’S brings you complete explanation on Binomial Theorem along with solved examples. BYJU’S uses a different approach altogether that involves video classes and one to one mentoring so that you can understand the difficult concepts of math easily. More emphasis is given in understanding of the concept other than completion of the Math Syllabus, to help the students ace in later stages of life.

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