If | array lll a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 array |=0 and vectors (1, a, a^2 ) (1, b, b^2 ) and (1, c, c^2 ) are non-coplanar, then the product a b c is _____

Step-by-step solution

We can rewrite, aligned & | array lll a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 array |= | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array |+ | array lll a & a^2 & a^3 b & b^2 & b^3 c & c^2 & c^3 array |=0 & = | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array |+a b c | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array |=0 & =(1+a b c) | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array |=0 & =(1+a b c)=0 [ | array lll a & a^2 & 1 b & b^2 & 1 c & c^2 & 1 array | 0 ] & a b c=-1 aligned