The term independent of x(x>0, x 1) in the expansion of [ (x+1) (x^ 2 / 3 -x^ 1 / 3 +1 ) - (x-1) (x- x ) ]¹⁰

Step-by-step solution

aligned & [ (x+1) (x^ 2 / 3 -x^ 1 / 3 +1 ) - (x-1) (x- x ) ]¹⁰ & = [ (x^ 1 / 3 )^3+1^3 (x^ 2 / 3 -x^ 1 / 3 +1 ) - ( x )^2-1 x ( x -1) ]¹⁰ & = [ (x^ 1 / 3 +1 ) (x^ 2 / 3 -x^ 1 / 3 +1 ) (x^ 2 / 3 -x^ 1 / 3 +1 ) - ( x )^2-1 x ( x -1) ]¹⁰ & = [ (x^ 1 / 3 +1 )- ( x +1) x ]¹⁰= (x^ 1 / 3 -x^ -1 / 2 )¹⁰ aligned The general term is aligned T_ r+1 & ¹⁰ C_r (x^ 1 / 3 )^ 10-r (-x^ -1 / 2 )^r & = ¹⁰ C_r(-1)^r x^ 10-r 3 - r 2 aligned For the term independent of x , put array cc & 10-r 3 - r 2 =0 & 20-2 r-3 r=0 & 20=5 r r=4 & T₅=