Binomial Theorem For Positive Integral Indices

Binomial Theorem

Before understanding about Binomial Theorem, let us focus on what we have learnt,

We all know the expansion of \( (a+b)^2, (a+b)^3, (a-b)^2, (a-b)^3\)

\((a+b)^2\)=\(a^2+2ab+b^2\)

\((a+b)^3\)=\(a^3+3a^2 b+3ab^2+b^3\)

(\(a-b)^2\)=\(a^2-2ab+b^2\)

\((a-b)^3\)=\(a^3-3a^2 b+3ab^2-b^3\)

Using the above expansions, we can easily find out the values of,

\((105)^2\)=\((100+5)^2\)

=\(100^2+2×100×5+5^2\)

=\(10000 + 1000 + 25\) = \( 11025\)

Similarly,

\((101)^3\)=\((100+1)^3\)

=\( 100^3 + 3 × 100^2 × 1 + 3 × 100 × 1^2 + 1^3\)

\(1000000 + 30000 + 300 + 1\) = \( 1030301 \)

But, finding out the values of \((102)^6\), \((99)^5 \) with repeated multiplication is difficult. This is made easy with a theorem known as binomial theorem.

By using this theorem, we can expand \((a+b)^n\), where n can be a rational number. Binomial theorem for positive integral indices is discussed here.

Let us write the expansion of \((a+b)^n \) , [0≤n≤5 and n is an integer] and find the properties of binomial expansion.

\((a+b)^0\)=\(1\)

\((a+b)^1\)=\((a+b)\)

\((a+b)^2\)=\(a^2+2ab+b^2\)

\((a+b)^3\)=\(a^3+3a^2 b+3ab^2+b^3\)

\((a+b)^4\)=\(a^4+4a^3 b+6a^2 b^2+4ab^3+b^4\)

\((a+b)^5\)=\(a^5+5a^4 b+10a^3 b^2+10a^2 b^3+5a^4 b+b^5\)

Properties of the expansion:

  • If you notice the power of a and b, exponent of a [first quantity] is decreasing by 1 in the successive terms. Meanwhile, exponent of b is increasing by 1 in the successive terms.
    For example; in the expansion of

\( (a+b)^3 \) = \( a^3+3a^2 b+3ab^2+b^3\)In the first term \( a^3\) , exponent of a is 3 and exponent of b is 0.In the second term3\(a^2\) b, exponent of a is 2 and exponent of b is 1.In the third term \(3ab^2\), exponent of a is 1 and exponent of b is 2.In the fourth term \(b^3\), exponent of a is 0 and exponent of b is 3.

  • The total number of terms in the expansion is one more than the exponent or index of (a+b).For example; in the expansion of\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\)\((a+b)^4\)=\(a^4+4a^3 b+6a^2 b^2+4ab^3+b^4\)Index of (a+b) is 4 and the number of terms in the expansion is 5.
  • The exponent or index of (a+b) will be equal to the sum of exponents of a and b in each term of the expansion.

We will see the relation between index of (a+b)and the coefficients of the terms in the expansion.

  Index

Coefficients

0                                                                                           1

1                                                                              1                        1

2                                                                  1                        2                       1

3                                                      1                        3                      3                                1

4                                         1                        4                       6                          4                            1

5                            1                       5                    10                         10                          5                               1

&;

  • The coefficients of the first term and last term of the expansion \((a+b)^n\) [where n can be any integer] is 1
  • Adding the 1’s of the index 1 gives you the 2 for the index 2.
    Similarly, adding 1 and 2 of the index 2 gives you the 3’s of the index 3.

Refer the following figure for better understanding.

Binomial Theorem

The discussion above is a brief introduction to binomial theorem for positive integral indices. To learn more about binomial expansion, log onto www.byjus.com and fall in love with BYJU’s way of learning.’

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