Algebra II

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?

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  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

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  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

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