5

List a complete list of representatives of F[x]/ (p(x)) where F = Zz and p(x) = x*+x2 + 1....

Question

List a complete list of representatives of F[x]/ (p(x)) where F = Zz and p(x) = x*+x2 + 1.

List a complete list of representatives of F[x]/ (p(x)) where F = Zz and p(x) = x*+x2 + 1.



Answers

Determine all polynomials $P(x)$ such that $P\left(x^{2}+1\right)=(P(x))^{2}+1$ and $P(0)=0$

All right, so we're looking for some components, and this is question number 39 from 39. We really only have two components. We got one going this way, which is not, strangely enough, the X direction. And ah, we have the X and Y directions like this. This is the wider action. And this is the extraction 45 degrees. Which, by the way, would make this 45 degrees also, um, and then we have F two, but F two. I would prefer to draw this way because I like to draw them tail the tail. Um, let's think about what's going on here. Um, considering where 30 degrees is on the drawing, it would be this. And that's going to be f Whoops. F two, which is 150 Newtons, if that makes sense. Okay, So, components, um, let's see here, F one is going to be 200 times the sign of 45 degrees, which is just the square root of two to over to, um in the eye direction and the same thing in the J direction. Okay. F two is going to be well, the in the X direction is gonna be the co side of 30 degrees, and the co sign of 30 degrees is a square to three over two. So this is just going to be ah, 150. Whoops. Negative. 150 cause going in the negative direction. And, um, the sign of 30 degrees is 1/2 the J direction. All right, so simplifying these this one is just going to be 100 root. Teoh. Hi. 100 route to J F two is going to be negative. 75 Route three. I comma 75 j.

If we have the function after X equals negative X squared minus X cubed plus 11 x in the 1st 1 of kind of negative one will plug negative one in for X. Any evaluate and you could go right to a calculator as you see it here. Or we can analyze Negative one squared is one and then the naked. But that is negative. One negative one cubed is negative. Born in the negative of that is positive and then 11 times negative one is negative. 11 negative one plus 10 so half of negative one is negative. 11. Party we want to find effort to so we substitute Chew in for acts can evaluate again. You can go to the calculator as you see it here, or we can analyze to Spirit is four, and the negative of that is negative for to Cuba's eight. So it's minus Kate and then plus 11 times two is 22. Negative for minus eight plus 22 is 10 so f of two equals tens

F composed G off X is equal to half off and g off x now by putting that you are de ex Levi, get half off X plus one and they went by one minus X. Now, from here, we can i d does X plus one, they want to buy one minus X. Do you want your by X plus one new one by one minus X plus one minus one. Now from here we can write the equation s after simplification to x the wildlife one minus x multiplied by one minus X. They want by two. Now, after simply vacation, we will get f composed. G off X is equal to x no de compose half off X is equal Do g off half off x now by putting the will you off ffx Really that g uh, X minus one they wanted by X plus one. Now from here we can I d tasks Eggs black minus one. They want to buy X plus one plus one. Do you want to buy? Okay, one minus X minus one the wide by X plus one. Now after simplification, we can write it as two x divided by X plus one multiplied by X plus one. Do you want to buy two From here, we can light it asked G combos half off X is equal to thanks.

So for this one, I'm gonna do f of G of X. Where I'm gonna take this entire G and plug it in right here. That should give me 1/1 of her ex, which is just X Fergie of f of X. I'm gonna take it this entire f and plug it in for that X. It's what you give me 1/1 over X, which is


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