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Synthetic Division: Computations w/ Radicals

  • Given that x = -3 + sqrt(11) is a zero of x4 + 6x3 7x2 30x + 10,
    fully solve the equation
    x4 + 6x3 7x2 30x + 10 = 0.
  • You need to be very careful working problems like this. It's easy to make mistakes.

    Do the first couple of steps:


    The first couple steps...



    Multiply to get the entry that goes under the 7:


    (3 + sqrt(11))(-3 + sqrt(11)) = -9 + 11 = 2


    Do the next couple of steps:


    The next couple steps...


    Multiply to get the entry that goes under the 10:


    (-15 - 5sqrt(11))(-3 + sqrt(11)) =45 - 55 = -10


    Complete the division:


    completed division


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


    completed division

Note that, since this second synthetic division handled the conjugate radical 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 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/ Radicals." Purplemath. Available from Accessed


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