coprime polynomials

I keep getting the wrong answer...

Show that the polynomials

f(x) = x^4 + 3x^3 - 2x^2 - x - 1
g(x) = 2x^2 + 2x + 1

are coprime.

Thanks.

conside g(x)=0
x=-2+sqrt(4-8)/4
so over the reals g(x) is irreducible.
since g(x) is not a factor of f(x), as shown in your post, the result follows.

Isn't the hypothetical common factor allowed to be complex?

I've already got that

x^4 + 3x^3 - 2x^2 - x - 1 = (0.5x^2 + x - 2.25)*(2x^2 + 2x + 1) + (2.5x + 1.25)

I've been trying to use Euclid's Algorithm, but I'm not really sure where I'm going wrong (although I clearly am since I can't the highest common factor = 1).

Thanks.

You've only applied what's called the division algorithm once. The Euclidean algorithm requires you to keep going till you get 0, and then the previous term is the hcf.

So divide 2.5x+1.25 into 2x^2 + 2x + 1. See what remainder you get.

Then if it's non-zero divide the remainder into 2.5x+1.25 etc.

thanks for your help

I don't suppose you could help me with finding a,b (polynomials) s.t.

a(x)f(x) + b(x)g(x) = 1?