# Math Help - Constructing a polynomial -relatively simple

1. ## 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.

2. 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.
$f(x) = x(x-1)(x-2)(x-3)(x-4)$

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

lol hacker, give me another one please

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

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.

4. Originally Posted by hercules
lol hacker, give me another one please
f(x) = 0

5. Originally Posted by ThePerfectHacker
f(x) = 0

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.

6. 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.

7. 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.