# Math Help - GCD of two polynomials

1. ## GCD of two polynomials

My task is to find the gcd of $nx^{n+1}-(n+1)x+1$ and $x^{n}-nx-1$.

Long division isn't giving me a clue here. My hunch is that the gcd is one, but I'm not sure I can prove that by showing they are irreducible (over what ring would I do that anyway? the question does not specify) or something like that.

2. You can use the fact that $\gcd(a,b)$ will divide $ma+nb$, where $a,b,m,n$ are elements from your ring.

Applying this, $\gcd(nx^{n+1}-(n+1)x+1, x^{n}-nx-1)$ also divides $[nx^{n+1}-(n+1)x+1]-nx[x^{n}-nx-1]=-(n+1)x+1+n^2x^2+nx=n^2x^2-x+1$

You can try to draw a conclusion from this.

3. ## One more step?

If n=2, you can repeat the subtraction:
$(n^2x^2-x+1) - n^2(x^n-nx-1) =$
$(4x^2-x+1) - 4(x^2-2x-1) =$
$7x+5$