$$4x^4-16x^3+115x^2+4x-29=0$$
has roots: $$\alpha,\beta,\gamma,\delta$$
Show that $2-5i$ is a root of the equation.
This to me was simply sub $f(2-5i)$ and see if it equaled zero, which it did. However since this had to be done by hand, it took a while and was rather tedious for only 3 marks. I was wondering if there was any other way this could be done out of pure curiosity, or I can say it's 'just of of those questions'.
Complex roots have to appear with their conjugates. So $2+5i$ also has to be a root. The quadratic polynomial for these 2 roots is $x^2-4x+29$. So you can check to see whether this quadratic is a factor of the given polynomial, either by determining other factors or direct division.