I got stuck on this question: Find the monic gcd of $f(x)=x^5-6x^4+13x^3-11x^2+x+5$ and $g(x)=x^2-3x+2$.
I worked through the Euclidean algorithm, first multiplying $g(x)$ with $x^3$ but then the remainder term has a larger power than $g(x)$. I have attached the working.
I know I am probably making a stupid mistake but can someone let me know how to get around this problem?
Consider factoring $g(x)$. By inspection, $$g(x) = x^2 - 3x + 2 = (x -1)(x-2)$$ Now check if either $(x-1)$ or $(x-2)$ is a factor of $f(x)$. Clearly, $x - 2$ cannot be a factor of $f(x)$. Why not?