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Synthetic Division: Computations w/ Complexes

  • Given that 2 i is a zero of x5 6x4 + 11x3 x2 14x + 5,
    fully solve the equation  
    x5 6x4 + 11x3 x2 14x + 5 = 0.
  • Work carefully, keeping in mind the properties of complex numbers. In particular, remember that i2 = 1.

    Do the first couple steps:


     first steps


    Multiply to get the entry that goes below the


     (-4 - i)(2 - i) = -9 + 2i


    Add down: 


     Add down: 11 + (-9 + 2i) = 2 + 2i


    Multiply to get the entry that goes below the 1:


     (2 + 2i)(2 - i) = 6 + 2i


    Add down:


     Add down:  -1 + (6 + 2i) = 5 + 2i


    Multiply to get the entry that goes below the 14: 


     (5 + 2i)(2 - i) = 12 - i


    Add down: 


     Add down:  -14 + (12 - i) = -2 - i


    Multiply to get the entry that goes below the 5: 


     (-2 - i)(2 - i) = -5


    Add down: 


     completed division


    Now you're ready for the next division, which works out like this: 


     completed division


Note that, since since this second synthetic division handled the conjugate complex root, all the complex coefficients disappeared. You should expect this to happen. Whenever you have roots that are conjugates, dividing out one of those roots will make things very messy, but then dividing out the other root will clean things back up. Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

If you want to get the right answers, do not try to do the messier parts in your head or in the margins; take out a sheet of scratch paper and do your work properly.

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Cite this article as:

Stapel, Elizabeth. "Synthetic Division: Computations w/ Complexes." Purplemath. Available from Accessed


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