Explain me, please

Question

Explain me, please

loser66 · Answer

My questions are
Why do we check just P(0),P(1), P(-1) , P(2), P(-2) only while $Z_5=\{0,\pm1,\pm2,\pm3,\pm4\}$

loser66 · Answer

In part b) apparently, P(1) =1, P(-1) =-1, Why they conclude that P(x) has no roots in Z5?

loser66 · Answer

attachment

loser66 · Answer

@ganeshie8

ganeshie8 · Answer

what values do you want to check ?

ganeshie8 · Answer

{-2, -1, 0, 1, 2}  give you all residues in mod 5 right ?

ganeshie8 · Answer

notice that  $\large -2 \equiv 3$ and   $\large -1 \equiv 4$

loser66 · Answer

Oh, yes, Got this part. How about part b?

tkhunny · Answer

Can't open a docx, sorry.

ganeshie8 · Answer

i cant open that file too, could you take a pic/pdf and post it

loser66 · Answer

ganeshie8 · Answer

yeah none of them are congruent to  $\large 0 \pmod 5$ so it is a prime polynomial

ganeshie8 · Answer

not reddicible into simple factors in z5

ganeshie8 · Answer

By factor theorem we have :
$$\large \text{P(k) = 0} \iff  \text{x-k  is a factor of P(x)}$$

loser66 · Answer

|dw:1411507596009:dw|

loser66 · Answer

Or we just consider integers?

ganeshie8 · Answer

it will definitely have 3 zeroes and atleast 1 real root for sure
but that real root is NOT an integer

ganeshie8 · Answer

yes we are considering integer roots modulo 5

loser66 · Answer

Got it. Thanks a ton...:)