Finding the greatest common divisior

[stan]

I got a bit stuck with finding the gcd for [MATH]x^5+x^3+x^2+x+2[/MATH] and [MATH]x^4+x^2+1[/MATH].
I started with the Eucledian algorithm and got that
[MATH]x^5+x^3+x^2+x+2 = (x^4+x^2+1)\cdot x+(x^2+2)[/MATH][MATH]x^4+x^2+1=(x^2+2)\cdot (x^2-1)+3[/MATH]But then got stuck..
Should the next step be [MATH]x^2+2=3\cdot \frac{2}{3}+ x^2[/MATH] or how do proceed?

 

[Dr.Peterson]

I got a bit stuck with finding the gcd for [MATH]x^5+x^3+x^2+x+2[/MATH] and [MATH]x^4+x^2+1[/MATH].
I started with the Eucledian algorithm and got that
[MATH]x^5+x^3+x^2+x+2 = (x^4+x^2+1)\cdot x+(x^2+2)[/MATH][MATH]x^4+x^2+1=(x^2+2)\cdot (x^2-1)+3[/MATH]But then got stuck..
Should the next step be [MATH]x^2+2=3\cdot \frac{2}{3}+ x^2[/MATH] or how do proceed?

Not quite. When you divide by 3 (a polynomial of degree 0), the remainder has to have lower degree! The result of this last division is quotient [MATH]\frac{1}{3}x^2+\frac{2}{3}[/MATH] and remainder 0.

So the final non-zero remainder was 3, and that is the GCD; you can call it, instead, 1, since constant factors can be ignored.