Are there $a(x),b(x) \in \mathbb{Q}[x]$ such that $a(x)f(x)+b(x)g(x) = x^2 + 1$
Where $f(x) = x^4+4x^3-7x+2$ and $g(x)=x^2+3x-4$
I have no idea how to approach this problem. I tried to visualize it as one with integers, but there seems to be some missing info - one function with two unknowns. Is there a special property of polynomials I'm missing here?
Well, for any polynomials $f(x),g(x)$ with coefficients in a field like the rational numbers, one can compute their greatest common divisor (gcd) $d(x)$ and represent it as a linear combination of these polynomials by using the extended Euclidean algorithm, i.e.
$f(x)s(x) + g(x)t(x) = d(x).$
In your example, you have to check if the right-hand side is the gcd or a multiple of it (and so lies in the ideal generated by $f(x)$ and $g(x)$), or it is not in the ideal (then the equation has no solutions $s(x)$ and $t(x)$).