Polynomial Division Homework: Q[x], Z5[x], R[x]

Click For Summary

Homework Help Overview

The problem involves dividing the polynomial 2x5 + x - 1 by 3x2 + 1 across different polynomial rings: Q[x], Z5[x], and R[x]. Participants are exploring the implications of polynomial division in these contexts.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss their attempts at polynomial long division, noting differences in results across the various polynomial rings. There is confusion regarding the handling of coefficients in Z5, particularly with respect to inverses and modular arithmetic.

Discussion Status

Some participants have provided critiques of each other's work, questioning the correctness of division steps and the treatment of coefficients. There is ongoing exploration of how to properly represent fractions in Z5, with some suggesting alternative approaches to the division process.

Contextual Notes

Participants are grappling with the rules of polynomial division in different fields, particularly the limitations of using fractions in Z5. There is also mention of specific calculations that may not align with standard practices in Q[x] and R[x].

sarah77
Messages
27
Reaction score
0

Homework Statement



Divide 2x5+x-1 by 3x2+1 in Q[x], Z5[x], and R[x]

The Attempt at a Solution



I believe the answer should be the same in Q[x] and R[x] and after division
I got 2/3x3-2/9x with remainder 1 2/9x-1. I had trouble in Z5 and would like someone to help me.
So far I have 3-1x3-1-1x but I do not think I am on the right track.

Please see attached work that I have completed so far..
 

Attachments

Last edited:
Physics news on Phys.org
I have attached the work I have done so far, please critique and help me finish. I am confused on what -3^-1x^3 becomes in Z5. Does it become 3x^3?
 
Hmm, I don't think your long division was correct. What do you want to multiply 3x2 by to get the 2x5 term in Z5. Essentially, what times 3 equals 2? That should be the first step to that division.
 
I have attached my work to this reply. Is this correct?
 

Attachments

Has anyone been able to read my work and critique it? Please let me know if I am on the right track..
 
\frac{2x^5}{3x^2}=\frac{2}{3}\,x^3\,, not 4x3

Added in Edit:

Unless you're doing Z5[x], in which case it is 4x3, since 12 ≡ 2 mod 5 .
 
Last edited:
How does 3 go into 4? Why couldn't you do a fraction decomposition?
 
flyingpig said:
How does 3 go into 4? Why couldn't you do a fraction decomposition?

It is in Z5 so 3x4=12 which is 2 in Z5 or am I missing something?

I cannot use fractions, I don't think 2/3 is in Z5
 
SammyS said:
\frac{2x^5}{3x^2}=\frac{2}{3}\,x^3\,, not 4x3


It is in Z5, so I cannot use 2/3, it would be 3-1 and I tried that and it was incorrect..
 
  • #10
SammyS said:
\frac{2x^5}{3x^2}=\frac{2}{3}\,x^3\,, not 4x3

Added in Edit:

Unless you're doing Z5[x], in which case it is 4x3, since 12 ≡ 2 mod 5 .


ok, so the division was correct then? All of it? Thank you for checking my work!
 
  • #11
sarah77 said:
It is in Z5 so 3x4=12 which is 2 in Z5 or am I missing something?

I cannot use fractions, I don't think 2/3 is in Z5

Right, I missed that your pdf was for Z5[x] .

It looks good.

Check by taking: (3x2+1)(4x3+2x) + 4x+4 .

It works out.
 
  • #12
Ok, thank you so much!
 

Similar threads

  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
5K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
3
Views
3K
  • · Replies 24 ·
Replies
24
Views
4K
  • · Replies 5 ·
Replies
5
Views
2K
Replies
7
Views
2K
  • · Replies 4 ·
Replies
4
Views
4K