absolute proofs. – C.P. STEINMETZ v . Introduction In earlier classes, we have learnt how to find the squares and cubes of binomials like a + b and a – b .
Using them, we could evaluate the numerical values of numbers like ( ) = ( – ) , ( ) = ( – ) , etc. However, for higher powers like ( ) , ( ) , etc., the calculations become difficult by using repeated multiplication. This difficulty was overcome by a theorem known as binomial theorem. It gives an easier way to expand ( a + b ) n , where n is an integer or a rational number.
In this Chapter, we study binomial theorem for positive integral indices only. . Binomial Theorem for Positive Integral Indices Let us have a look at the following identities done earlier: ( a+ b ) = a + b ≠ ( a+ b ) = a + b ( a+ b ) = a + ab + b ( a+ b ) = a + a b + ab + b ( a+ b ) = ( a + b ) ( a + b ) = a + a b + a b + ab + b In these expansions, we observe that The total number of terms in the expansion is one more than the index. For example, in the expansion of ( a + b ) , number of terms is whereas the index of ( a + b ) is .
Powers of the first quantity ‘ a ’ go on decreasing by whereas the powers of the second quantity ‘ b ’ increase by , in the successive terms. (iii) In each term of the expansion, the sum of the indices of a and b is the same and is equal to the index of a + b .