# Constructing a polynomial -relatively simple

• May 5th 2008, 07:17 PM
hercules
Constructing a polynomial -relatively simple
Construct a single polynomial f(x) in Z5[x] (the set of all polynomials in Z mod 5) such that every element of Z5 = {[0],[1],[2],[3],[4]} is a root of f(x). (Z is the set of integers.)

Please tell how you can quickly construct a polynomial with such properties
Thank you.
• May 5th 2008, 07:20 PM
ThePerfectHacker
Quote:

Originally Posted by hercules
Construct a single polynomial f(x) in Z5[x] (the set of all polynomials in Z mod 5) such that every element of Z5 = {[0],[1],[2],[3],[4]} is a root of f(x). (Z is the set of integers.)

Please tell how you can quickly construct a polynomial with such properties
Thank you.

$f(x) = x(x-1)(x-2)(x-3)(x-4)$ (Rofl)
• May 5th 2008, 07:21 PM
hercules
Quote:

Originally Posted by ThePerfectHacker
$f(x) = x(x-1)(x-2)(x-3)(x-4)$ (Rofl)

lol hacker, give me another one please

Quote:

Originally Posted by ThePerfectHacker
$f(x) = x(x-1)(x-2)(x-3)(x-4)$ (Rofl)

I never thought looking at that smilie was this annoying. And thanks for the answer. I was trying to figure if possible what other polynomials met the condition and an easy way for that....wasting my brain.
• May 5th 2008, 07:32 PM
ThePerfectHacker
Quote:

Originally Posted by hercules
lol hacker, give me another one please

(Rofl) f(x) = 0 (Rofl)
• May 5th 2008, 07:38 PM
hercules
Quote:

Originally Posted by ThePerfectHacker
(Rofl) f(x) = 0 (Rofl)

hey a polynomial needs more than one term....i can't believe your giving the answer but completely dodging the answers i'm hoping for-the reason i made this thread. But good ones. ....No more smilies please....nightmares.
• May 5th 2008, 07:44 PM
ThePerfectHacker
Quote:

Originally Posted by hercules
hey a polynomial needs more than one term....i can't believe your giving the answer but completely dodging the answers i'm hoping for-the reason i made this thread. But good ones. ....No more smilies please....nightmares.

The zero polynomial is still a polynomial.

For any polynomial $f(x)$ the polynomial $g(x) = x(x-1)(x-2)(x-3)(x-4)f(x)$ will have the desired properties.
Furthermore, any other such polynomial which contains zeros of $0,1,2,3,4$ must have this form.
• May 5th 2008, 07:54 PM
hercules
Quote:

Originally Posted by ThePerfectHacker
The zero polynomial is still a polynomial.

For any polynomial $f(x)$ the polynomial $g(x) = x(x-1)(x-2)(x-3)(x-4)f(x)$ will have the desired properties.
Furthermore, any other such polynomial which contains zeros of $0,1,2,3,4$ must have this form.

Thank you ...needed to clear my misconceptions.