find the term independent of x in binomial expression

(x4+1/x)10

(3x2-1/3x)9

(x-1/x2)3n

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.