IIT JEE , AIEEE Forum - Mathematics , Algebra

elastiboysai

1. Write the following polynomial as a product of irreducible polynomials in Z[X]

f(X) = X^2005 ~ 2005X + 2004.

elastiboysai

ok!

lemme see who can do it.

sivaramakh

gr8 gob

hsbhatt

x²⁰⁰⁵-2005x+2004 = x²⁰⁰⁵-1 - 2005x+2005 = x²⁰⁰⁵-1 - 2005(x-1)

This can be factorised as x²⁰⁰⁵-1 = (x-1) (x²⁰⁰⁴+x²⁰⁰³+..+x²+x+1)

Hence the required factorisation is (x-1) (x²⁰⁰⁴+x²⁰⁰³+..+x²+x-2004)

Now, (x²⁰⁰⁴+x²⁰⁰³+..+x²+x-2004) becomes zero for x =1 and hence

(x²⁰⁰⁴+x²⁰⁰³+..+x²+x-2004) = (x-1) P(x)

It remains to prove that (x²⁰⁰⁴+x²⁰⁰³+..+x²+x-2004) has no more integer zeros.

Which is where I am stuck

elastiboysai

edited

sorry

i din read ur post completely,(the 2nd n 3rd last lines)

hsbhatt

I have not written it explicitly, but in my previous post I have reduced the poly given to the form (x-1)² P(x). Now does P(x) have any further integer roots?

konichiwa2x

yeah..getting only this far .

Are you sure it can be reduced further?

edit: mistake corrected.

elastiboysai

Guyz,

look at the gn fn

f(-2)<0

f(-1)>0 so ders a root between -2 and -1.

also, f'(x)>0 for all x>1

so the roots of dis must be<=1

hsbhatt

Yeah, but integers exist that are less than 1 too

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