Question

determimant y+z     x    y  ..=(x+y+z)(x-z)2

Answer 1

Whenever two of the variables are equal, two of the rows are equal, so the determinant is zero. Thus the determinant must contain each of the three differences of the variables as linear factors. Since the determinant is a cubic polynomial, it must be of the form k(x−y)(y−z)(z−x) with k∈R, and comparing coefficients e.g. of xy2 yields k=1.