Can I still use the rational root theorem if the remaining roots are imaginary?

Answer 1

The rational root theorem can only find rational roots BUT the results of finding a rational root will leave a quotient that is of smaller degree that you can sometimes use other methods on to find the complex roots. Example: x³ - 3x² + x - 3

The rational roots theorem says that if there are any rational roots, they will be +1 or +3. You try synthetic division and determine that x = 3 is a root but in so doing you find the quotient of

(x³ - 3x² + x - 3)/(x - 3) is x² + 1. Now you can set that equal to zero and find the other roots (complex) are + i