Find all the roots of the equation x4 - 2x3 + 14x2 - 18x + 45 = 0 given that 1 + 2i is one of its roots.

Question

Find all the roots of the equation x4 - 2x3 + 14x2 - 18x + 45 = 0 given that 1 + 2i is one of its roots.

Answer 1

If one root is 1 + 2i, the other root must be 1 - 2i.

 

So (x -1 -2i)(x-1 +2i) = (x-1)² + 4 = x² - 2x +5 must be a factor of the original equation

 

Using long division, you will find that

x⁴ - 2x³ + 14x² -18x + 45 = (x² - 2x + 5)(x² + 9) = 0

 

I think you should be able to get all the roots from here.