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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

find the term independent of x in binomial expression


(x4+1/x)10


(3x2-1/3x)9


(x-1/x2)3n

Grade:

1 Answers

Aman Bansal
592 Points
11 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)?
 
  1. The number of terms in the expansion is one more than the exponent.

  2. In each expansion:
  1. The exponent of the first term is same as the exponent of the binomial. The exponent of y in the first term is zero. 

  2. Subsequently, in each successive term, the exponent of x decreases by 1 with a simultaneous increase of 1 in the exponent of y. 

  3. The sum of the exponents of x and y in each term is equal to the exponent of the binomial. 

  4. 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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Thanks

Aman Bansal

Askiitian Expert

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