Perform the Indicated operation

(x+1)^3 in Z3[x]

Question

Perform the Indicated operation

(x+1)^3 in Z3[x]

anonymous · Answer

$$ (x+1)^{3}  in   Z_{3}[x]$$

terenzreignz · Answer

$$\huge (x+1)^3 \ \ \ \ in \ \mathbb{Z}_3[x]$$
LOL any idea how to do this? :)

anonymous · Answer

do i just use f(0) and f(1) to solve for them in Z3? I'm not sure really thats why I'm asking and my book doesn't even address it.

anonymous · Answer

hint $3\equiv 0$

terenzreignz · Answer

Freshman's dream :D
Anyway, evaluate this
(x+1)^3 normally first, and tell me what you get.

anonymous · Answer

lol

terenzreignz · Answer

Cube of a binomial?

anonymous · Answer

1+3 x+3 x^2+x^3

terenzreignz · Answer

That's right :)
$$\huge x^3 + 3x^2+3x+1$$but... we take everything in mod 3...
In particular, in 
$$\large \mathbb{Z}_3$$
3=0, right?

anonymous · Answer

yes

anonymous · Answer

so is it X^3 + 1?

terenzreignz · Answer

You got it :)
And now, for any prime number p

terenzreignz · Answer

$$\huge (a+b)^p=a^p+b^p \ \ \ \ \ in \ \mathbb{Z}_p$$

anonymous · Answer

i always suspected my math teachers were lying, and $(a+b)^3=a^3+b^3$

terenzreignz · Answer

Oh abstract algebra... so deceptive :D
That was fun... but I'm getting drowsy :/
-----------------------------------------
Terence out

anonymous · Answer

Thank you