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# find the term independent of x in binomial expression(x4+1/x)10(3x2-1/3x)9(x-1/x2)3n

9 years ago

Dear Deepika,

 We know that any algebraic expression with two variables is called a binomial. Observe the following powers of the binomial (x + y).     (x + y)0 = 1      (x + y)1 = x + y      (x + y)2 = x2 + 2xy + y2      (x + y)3 = x3 + 3x2y + 3xy2 + y3      (x + y)4 = (x + y)3(x + y)                  = (x3 + 3x2y + 3xy2 + y3) (x + y)                  = x4 + 3x3y + 3x2y2 + xy3 + x3y + 3x2y2 + 3xy3+ y4     (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 What do you notice in the above expansion of various powers of (x + y)? The number of terms in the expansion is one more than the exponent. In each expansion: The exponent of the first term is same as the exponent of the binomial. The exponent of y in the first term is zero.  Subsequently, in each successive term, the exponent of x decreases by 1 with a simultaneous increase of 1 in the exponent of y.  The sum of the exponents of x and y in each term is equal to the exponent of the binomial.  The exponent of x in the last term is zero and that of y is equal to the exponent of the binomial.

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