Algebra II
posted by Nick on .
Solve x^4  12x  5 = 0 given that 1 + 2i is a root.
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?

If (1+2i) is a solution, its complex conjugate (12i) must be a solution.
In other words
x = 1+2i
x = 12i
are solutions
so
x+12iand 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^22x1
well
that is
(x1)(x1)
so my four solutions are those two complex conjugates and x=1 and x =1 again 
Whoa  a bit hasty there
x^22x1
can not factor that so easily!
I will have to solve it
x^22x1 = 0
x = (1/2)(2 +/sqrt(4+4))
=(1/2)(2 +/2sqrt(2))
x = 1+sqrt(2)
or
x = 1sqrt(2)
are the last two solutions