If the determinant | array ccc 2 x & ^2 x & 2 x ^2 x & 2 x & ^2 x 2 x & ^2 x & 2 x array | is expanded in powers of x , then the constant term in the expansion is

Step-by-step solution

Let = | array ccc 2 x & ^2 x & 2 x ^2 x & 2 x & ^2 x 2 x & ^2 x & 2 x array | On expanding along R₁ . aligned & = 2 x [ ^2 2 x- ^4 x ]- ^2 x & [ ^2 x 2 x- ^2 x 2 x ] & + 2 x [ ^2 x ^2 x- ^2 2 x ] & = 2 x [ (2 ^2 x-1 )^2- ^4 x ] & - ^2 x [ 2 x ( ^2 x- ^2 x ) ] & + 2 x [ (1- ^2 x ) ^2 x- (2 ^2 x-1 )^2 ] & = (2 ^2 x-1 ) [4 ^4 x+1-4 ^2 x- ^4 x ] & - (1- ^2 x )[ 2 x(- 2 x)] & + 2 x [ ^2 x- ^4 x-4 ^4 x-1+4 ^2 x ] & = (2 ^2 x-1 ) [3 ^4 x+1+5 ^2 x-5 ^4 x-1 ] & + (1- ^2 x ) [ (2 ^2 x-1 )^2 ] & (2 ^2 x-1 ) [5 ^2 x-2 ^4 x ]+