Need help in hard exercise

[Guest]

Settle, if there exist in pairs different rational numers, that polynomials
P(x) = [tex]x^{3}[/tex] + ax^{2} + bx +c and Q(x)= x^{3} +bx^{2} + cx + a
they have a common irrational element.

[Guest]

Settle, if there exist in pairs different rational numers, that polynomials
P(x) = [tex]x^{3}[/tex] + a[tex]x^{2}[/tex] + bx +c and Q(x)= [tex]x^{3}[/tex] +b[tex]x^{2}[/tex] + cx + a
they have a common irrational element.

[HallsofIvy]

What does "common irrational element" mean? Do you mean "have the same irrational number as root"?

[GeradHum]

To check if P(x) and Q(x) share a common irrational root, find their greatest common divisor (GCD). If the GCD is a polynomial with irrational roots, then yes, they share a common irrational root. Otherwise, no. You need to compare roots or use the resultant to see if they have common roots.