Using Determinant Identities to solve

[MMhawk607]

Sorry for the format, I'm on my phone.

Lets say the matrix is

| 1 1 1 |
| a b c |
| a^2 b^2 c^|

Or
{{1,1,1} , {a, b,c} , {a^2, b^2,c^2}}

Show that it equals to
(b-c)(c-a)(a-b)

I did the determinant and my answer was
(bc^2) - (ba^2) - (ac^2) + (ab^2) + (ca^2) - (cb^2)


This is the same thing as the answer but multiplied out and it is hard to just factor the answer into what we're supposed to prove. So is there any other way, I assume using the determinant identities, to easily get (b-c) (c-a) (a-b)
Or is multiplying them out and showing that it's the same as my Determinant enough.

 

[HallsofIvy]

Yes, just stating that (b-c)(c-a)(a-b)= (bc^2) - (ba^2) - (ac^2) + (ab^2) + (ca^2) - (cb^2) is sufficient?

 

[AlephZero]

You could say $$\left|\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{matrix}\right|
= \left|\begin{matrix} 1 & 0 & 0 \\ a & b-a & c-a \\ a^2 & b^2-a^2 & c^2-a^2 \end{matrix}\right| $$
$= (b-a)(c-a)(c+a) - (c-a)(b-a)(b+a)$. It's easy to get the answer given from there.