determinant-product of linear factors

Find det A as a product of linear factors, with A the 4x4 matrix:
{(1,1,1,1) (a,b,c,d) (a^2,b^2,c^2,d^2) (a^3,b^3,c^3,d^3)}

I've got so far:

(b-a)(d-a)(c-a)det{(1,0,0,0) (a,1,0,0) (a^2, b+a, c-b, d-b) (a^3, b^2+a^2, c^2-b^2, d^2-b^2)

Do I need to do anymore or is this correct?!

Reply 1

Neapolitan

Google Van der Monde Determinant.

Reply 2

Wrangler

Think what happens to the determinant of a matrix if any of its rows/columns are linearly dependant. Clearly, the determinant must have a factor of:

(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)

You'll then notice this is of the correct order in {a,b,c,d}, and a little thinking will allow you to conclude your work here is done!