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Factorise this determinant!

Apologies for the ... but it's the only way to space it out as a 3 x 3 matrix.

Q12 (iv) Ex 5B from the MEI FP2 text.

Factorise (imagine the determinant bars)

1........1.....1
x^2...y^2...z^2
yz.....xz.....xy

They get (x-y)(y-z)(z-x)(xy + yz + zx)

I can see the (x-y)(y-z)(z-x) from the cyclic nature of the thing, but where does the (xy + yz + zx) come from???

Much appreciated!

Reply 1

Totally Tom

I haven't done this yet, do you just find the determinant of this and then factorise it?

I've got the determinant.

Have you tried multiplying out the factorised determinant they have and checking?

I'm going to do it now, if we both do it chances are we would be right if both got same answer.

Reply 2

DFranklin

I don't think this can be right. The determinant is a degree 4 polynomial, but the answer given has degree 5.

Reply 3

Totally Tom

Yes, I just spent at least 10 minutes multiplying out those brackets to arrive at the same conclusion.

The brackets multiply out to be:

y³x²-x³y²+y²z³-y³z³-x²z³+x³z²

Whereas the actual determinant is:

y³x+z³y+x³z-z³x-x³y-y³z

Geez that took ages.

Reply 4

DFranklin

thomas795135

Yes, I just spent at least 10 minutes multiplying out those brackets to arrive at the same conclusion.Easier to factorize the determinant, at which point the error is fairly obvious.

Spoiler

Reply 5

Totally Tom

You see I did attempt to do that, but then I gave up as factorising a fourth degree polynomial with x y and z is quite hard for my level :frown:

.

Is there a method that you go through or is it 'just obvious'?

Reply 6

kalowski

DFranklin

I don't think this can be right. The determinant is a degree 4 polynomial, but the answer given has degree 5.

Exactly what I thought!
Thankfully, Ive got the answer now - same as yours -thanks for doing it.
I performed a couple of column operations and factorised then worked out the simpler determinant and got it