Solve x^4 - 12x - 5 = 0 given that -1 + 2i is a root.

Question

I’ve used synthetic division to get the answers x = 1 +/- 2i , 1 +/- sqrt 2

I’ve having trouble checking the answers though because of that imaginary number. How can I check these kind of problems?

Answer 1

If (-1+2i) is a solution, its complex conjugate (-1-2i) must be a solution.
In other words
x = -1+2i
x = -1-2i
are solutions
so
x+1-2iand x+1+2i are both factors
so their product should be a factor
I get x^2+2x+5
now do your division
I get
x^2-2x-1
well
that is
(x-1)(x-1)
so my four solutions are those two complex conjugates and x=1 and x =1 again

Answer 2

Whoa - a bit hasty there
x^2-2x-1
can not factor that so easily!
I will have to solve it
x^2-2x-1 = 0
x = (1/2)(2 +/-sqrt(4+4))
=(1/2)(2 +/-2sqrt(2))

x = 1+sqrt(2)
or
x = 1-sqrt(2)
are the last two solutions

Answer 3

Solve x^4 - 12x - 5 = 0 given that -1 + 2i is a root.