Question

Given that kx^3+2x^2+2x+3 and kx^3 -2x + 9 have a common factor,waht are the possible values of k?

Answer 1

if they have a common factor (kx-r) (root x=r/k) then
k(r/k)^3 + 2(r/k)^2 + 2(r/k) + 3 = k(r/k)^3 - 2(r/k) + 9 = 0
2(r/k)^2 + 4(r/k) - 6 = 0
(r/k)^2 + 2(r/k) - 3 = 0
r/k = -3 or 1

Now use the equations where f(r/k) = 0 to set a second condition. That will narrow down r and k.

Of course, it could be that k = a*b, in which case you could have factors (ax-p) and (bx-q) but I expect that is beyond what they want.