BINOMIAL THEOREM

BINOMIAL THEOREM BINOMIAL THEOREM We now arrange the coefficients in these expansions as follows (Fig . ): Do we observe any pattern in this table that will help us to write the next row? Yes we do. It can be seen that the addition of ’s in the row for index gives rise to in the row for index .

The addition of , and , in the row for index , gives rise to and in the row for index and so on. Also, is present at the beginning and at the end of each row. This can be continued till any index of our interest. We can extend the pattern given in Fig .

by writing a few more rows. Pascal’s Triangle The structure given in Fig . looks like a triangle with at the top vertex and running down the two slanting sides. This array of numbers is known as Pascal’s triangle , after the name of French mathematician Blaise Pascal.

It is also known as Meru Prastara by Pingla. Expansions for the higher powers of a binomial are also possible by using Pascal’s triangle. Let us expand ( x + y ) by using Pascal’s triangle. The row for index is Using this row and our observations (i), (ii) and (iii), we get ( x + y ) = ( x ) + ( x ) ( y ) + ( x ) ( y ) + ( x ) ( y ) + ( x )( y ) +( y ) = x + x y + x y + x y + xy + y .

Fig . Fig . MATHEMATICS Now, if we want to find the expansion of ( x + y ) , we are first required to get the row for index . This can be done by writing