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

[sarah77]

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..

 

[sarah77]

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?

 

[Sethric]

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.

 

[sarah77]

I have attached my work to this reply. Is this correct?

 

[sarah77]

Has anyone been able to read my work and critique it? Please let me know if I am on the right track..

 

[SammyS]

\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 .

 

[flyingpig]

How does 3 go into 4? Why couldn't you do a fraction decomposition?

 

[sarah77]

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

 

[sarah77]

\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..

 

[sarah77]

\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!

 

[SammyS]

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.

 

[sarah77]

Ok, thank you so much!