Complex numbers and polynomials

[ornate sparrow]

hello! i'm from Brazil and i'd like some help with this question. i don't have the answer, but i'd be grateful if someone could analyze my question development (it's in portuguese, but i'd like to know if the answer is correct), thanks!

[clever cape]

of course, that's not correct. You kind of did this (a+bx)(a+bx²)=(a+b)²(x·x²), which is definitely not true.

[clever cape]

1. P(x) can be evaluated in a number of ways: by using generating functions, or just by adding and subtracting 1, 1+x, 1+x+x^2, ... a number of times, or by differentiating f(x)=1+x+...+x^(N-1) two times, where N=2011.
2. To find the final product just use the fact that f(x)=(x-z)(x-z^2)...(x-z^(N-1)). Thus, you should get this answer

[ornate sparrow]

alright, i'm going to try this
thanks!

[ornate sparrow]

@clever cape i got this, but the answer is (1005^2011 - 1004^2011).2011^2009, but i found (1005^2011 - 1004^2011).2011^2008

i think that this is the correct idea, but i have to analyze my math calculations again

[clever cape]

One piece of advice: don't do this with numbers. Instead of a single letter N, you wrote four digits, thereby making understanding and verification four times more difficult, increasing the chance of error fourfold, and also multiplying the time and effort needed to write it all down. Besides, if you used just N, you could easily plug in a small value, say N=4, and quickly check by hand whether your formulas work correctly.

I don't know why you think the answer needs 2009, but 2008=2011-3 is the correct one.