(Original post by porkstein)
TheMagicMan is being almost deliberately confusing, introducing notation that simply isn't necessary for this question. He's essentially just expressing what the question says in words with symbols, presumably because that satisfies him or something. I can't see how he could've thought it would help you in any way.
It really isn't relevant, but in case you'd like to know what he means (it's not as hard as it looks):
just represents some general constant that is multiplied by the x4
term. You could call it "the x4
coefficient". As such the expression above involving that capital sigma (the big E-like symbol) is just a way of expressing some general polynomial (of degree n).
, which should really be written as
So looking back at the equation involving this capital pi, what we're trying to say is that the product of all the roots of the equation is equal to a0
, the constant term of the polynomial. This is similar to what I said about fgh being equal to 26. The (-1)^n bit just comes about because if, say, 5 is a root of the equation, then in the factorised form it's going to appear as (x minus
5). When all these roots are multiplied together you'll potentially get a minus term floating around, depending on how many things you multiply together.
This is just a fancy way of saying that the roots are all different. The second root of the equation, r2
, is not equal to the fifth root of the equation, r5
. Nor is the fifth root equal to the fourth root, and so on. The i just stands in for any number, as does j. When you run through all the combinations of roots, if you ask yourself, say, whether the third root of the equation equals the third root of the equation, the answer is obviously yes. Thus the i =/= j just says that the roots are all different, providing
you don't consider the same root twice in your comparisons.
This is just a fancy way of saying the roots are integers.