Polynomial division

Question

How can I know if $ \ \ 2(t-1) \ \ $ divides $ \ \ t^4+2t^3-2t^2-3t+2$ over $\mathbb Q$ ? I don't want to use the polynomial division algorithm.

My approach : $(t-1)$ divides the polynomial since $1$ is a zero of it. Now my doubt is if $2$ divides the polynomial. My idea was that asking this is equivalent to asking if $ 2| t^4-3t$ since the other coefficients of the polynomials are even.

Over $\mathbb{Q}$, of course $2$ divides any polynomial, since it's a unit. You have $t^4 - 3t = 2(\frac{1}{2}t^4 - \frac{3}{2}t)$. Over $\mathbb{Z}$, this isn't true, though. — Cocopuffs, Jul 29 '13 at 18:02

score 4 · Accepted Answer · answered Jul 29 '13 at 17:36

4

HINT:

$$t^4-3t=t^4-t^3+t^3-t^2+t^2-3t=t^3(t-1)+t^2(t-1)+t(t-1)+2t$$

Each term except the last is divisible by $t(t-1)$ which is even

answered Jul 29 '13 at 17:36

lab bhattacharjee

274,582

ok so if i understood right, we could say that $3$ divides $t^3-3t^2+5t$ since $t^3-3t^2+5t=t(t-1)(t-2)+3t$ right? – sigmatau Jul 29 '13 at 17:42
@AmireBendjeddou, try putting $t=3a-1,3a,3a+1$ – lab bhattacharjee Jul 29 '13 at 17:45
this is an example i invented at the moment i konw that..i was aksing if i'm right (: – sigmatau Jul 29 '13 at 17:46
1

@AmireBendjeddou, yes. you are. – lab bhattacharjee Jul 29 '13 at 17:47
How do you mean $t(t-1)$ is even? It's a polynomial.. – Cocopuffs Jul 29 '13 at 18:05
@Cocopuffs, $t$ is an integer, right? – lab bhattacharjee Jul 29 '13 at 18:08
@labbhattacharjee I doubt it, based on the title and tags – Cocopuffs Jul 29 '13 at 18:13
@Cocopuffs, if $t$ is not integer, does divisibility by $2$ make any sense? What do you think the nature of $t?$ – lab bhattacharjee Jul 29 '13 at 18:17
@labbhattacharjee Sure, you can define "divisibility" in any commutative ring – Cocopuffs Jul 29 '13 at 18:20

score 1 · Answer 2 · answered Jul 29 '13 at 17:48

1

Obviously $2$ divides $t^4-3t$

because $t^4-3t=(t^4-t)-2t=t(t^3-1)-2t=t(t-1)(t^2+t+1)-2t$

As $t(t-1)$ divisible by $2$ it implies that $t^4-3t$ divisible by $2$

answered Jul 29 '13 at 17:48

Siddhant Trivedi

2,298

score 1 · Answer 3 · answered Jul 29 '13 at 17:54

1

Clearly $t-1$ divides the polynomial (As you've already found why). Now note that $$t^4+2t^3-2t^2-3t+2\equiv t(t^3-1)\mod 2\equiv t(t-1) \mod 2\equiv0 \mod 2$$

answered Jul 29 '13 at 17:54

Samrat Mukhopadhyay

16,277

Polynomial division

3 Answers3