Find all the roots of the equation x^4 - 2x^3 + 14x^2 - 18x + 45 = 0 given that 1 + 2i is one of its roots.

Question

find all roots 

Answer 1

we know two roots because imaginary roots occur in conjugate pairs.

thus [x-(1+2i)][x-(1-2i)] = x²-2x-3 is a factor.

 

You can divide this quadratic factor out to get a cubic polynomial remaining factor, and you can also use the rational roots theorem thereafter.

 

Then all you have to do is to check your results.