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Binomial Theorem For Positive Integral Indices


 In algebra, the binomial theorem shows the algebraic expansion for binomial with powers. by using  binomial theorem it is possible to expand the power (x + y) n into a sum containing terms of the form axbyc, where the coefficient of every  term is a positive, and the sum of the exponents of x and y in each term is n.The expansion of positive integral indices is using binomial theorem and the explanation for binomial theorem is given in the following sections

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Binomial theorem:

With the help of this binomial theorem for positive integral index indices , we can expand any power of x + y into a sum of terms forming a polynomial. The terms are called as binomial co -efficients. Since it is only for positive integral indices,it expands only positive indices and not negative indices.

               (x+y) n =       nc0. Xn.y0 + nc1 .x (n-1) y1 + nc2 .x (n-2).y2+……..+nc (n-1) x1.y (n-1) + ncn.x0.yn

Where the corresponding binomial theorem coefficient example is in the form ‘nCk.

                                                  nCk = `(n!) / [k! (n-k)!]`

Example for expansion for positive integral indices:

Here are few examples for binomial theorem for positive integral indices,

Problem 1:

1. Expand (1+4x)6 using binomial theorem.

Binomial theorem [ (x+y) n =       nc0. Xn.y0 + nc1 .x (n-1) y1 + nc2 .x (n-2).y2+……..+nc (n-1) x1.y (n-1) + ncn.x0.yn     ]

Using theorem the positive integral indice can be expanded as,
    (1+4x)6 =   6C0 . 16. (4x)0+ 6C1 15 (4x)1+ 6C2 1 (4x)2+ 6C3 13 (4x)3+ 6C4 12 (4x)4+ 6C11 (4x)5  +   6C10 (4x)6    
                   = 1+ 5 (4x) +16x2 (10) + 64x(10) + 5 (256x) + 1024 x
                  = 1 + 20x + 160x2  + 640x3 + (640x) + 1024 x5               

Problem 2:

Expand (x+y)3 using binomail theorem,

Binomial theorem [ (x+y) n =       nc0. Xn.y0 + nc1 .x (n-1) y1 + nc2 .x (n-2).y2+……..+nc (n-1) x1.y (n-1) + ncn.x0.yn     ]

    (x+y)33C0 . x3. (y)0+ 3C1 x2 (y)1+ 3C2 x (y)2+ 3C3 x0 (y)3     

                  =  1 (x3) + 3 ( x2 (y)1 ) + 3 ( x (y)2 ) + 1 ((y)3)   

              =   x3 + 3  x2 y  + 3  x  y2 ) +  y3 


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Practice problems:

Expand the following using binomial theorem for positive integral indices:

1. Expand (`sqrt(2)` + 1)6 + (`sqrt(2)` - 1)6                       ============> [Answer: 198]

2. Expand (10.1)5   [Hint: use 10.1 = 10 +0.1]   ============> [Answer: 105101.00501]