Find the real numbers a,b such that the polynomials
$$p(x)=x^4-2x^3+3x^2+2x+a $$ $$q(x)=x^4-3x^3+4x^2+3x+b $$
have a common divisor of degree two.
My attempt: Euclid algorithm: we perform the divisions:
f:g obtain remainder r1 of degree 3
g:r1 obtain remainder r2 of degree 2
r1: r2 obtain remainder r3 of degree 1
And the condition must be that r3=0, but i get a not so nice system of equations with a and b
Is there any simpler method?
That's the method that I would use. But, in fact, doing it that way I got the system:$$\left\{\begin{array}{l}-2a^2+3ab+17a-b^2-13b=0\\a^2-2ab-11a+b^2+8b=5.\end{array}\right.$$In order to solve it, I used Mathematica, which gave me $3$ solutions: $(a,b)=(-4,-5)$, $(a,b)=(5,-5)$, and $(a,b)=(5,7)$. And they are indeed solutions of the original problem.