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Math Help - Need some help with finding zeros of a polynomial given one complex zero

  1. #1
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    Need some help with finding zeros of a polynomial given one complex zero

    Hi all! having some trouble with this question and have an exam in 3 days so hoping to get some help! leaving it late, I know.

    Anyway, so f(x)= x^4 - 3x^3+6x^2+2x-60 is the polynomial and 1+3i is the complex zero.

    So I used synthetic division to obtain this 1 4+3i 1+15i - 42+18i remainder -36-108i

    Then I used 1-3i, the conjugate which should also be a zero, yes? to obtain: 1 5 6 -36 and a remainder of -72 for my co-efficients. Which if the sign had been reversed I would have had an nice neat 0 remainder and I could have gone on from there but ive been over my working twice and cant find any mistakes I made- I am hoping that I have, as that means I don't have to learn how to deal with these remainders.

    Hoping someone can help me out here, thanks so much!
    Rhys
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  2. #2
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    Re: Need some help with finding zeros of a polynomial given one complex zero

    I find that synthetic division with complex zeros are a bit too tedious (for me) ...

    since the given complex zero and its conjugate are roots ...

    [x - (1+3i)][x - (1-3i)] = x^2 - 2x + 10

    ... the above quadratic is a factor of the original polynomial.

    \frac{x^4-3x^3+6x^2+2x-60}{x^2-2x+10} = x^2-x-6 = (x-3)(x+2)

    from which we can find the other two (real, in this case) roots.
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    Re: Need some help with finding zeros of a polynomial given one complex zero

    Quote Originally Posted by skeeter View Post
    I find that synthetic division with complex zeros are a bit too tedious (for me) ...

    since the given complex zero and its conjugate are roots ...

    [x - (1+3i)][x - (1-3i)] = x^2 - 2x + 10
    ... the above quadratic is a factor of the original polynomial.
    Ok thank you but I am unsure as to how you obtained the factor there, I can see how that quadratic is a factor but I dont know you obtained the complex factor from the original polynomial, if you might be able to explain the steps involved?
    Quote Originally Posted by skeeter View Post
    \frac{x^4-3x^3+6x^2+2x-60}{x^2-2x+10} = x^2-x-6 = (x-3)(x+2)
    So did you use log division here to obtain the result or some other method? Sorry I only really know the synthetic division method
    Thanks again!
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    Re: Need some help with finding zeros of a polynomial given one complex zero

    if z is a complex root, then so is its conjugate \bar{z} ... therefore (x - z) and (x-\bar{z}) are factors of the quartic.

    (x - z)(x-\bar{z}) =

     x^2 - zx - \bar{z} x + z \bar{z} =

    x^2 - (1+3i)x - (1-3i)x + (1+3i)(1-3i) =

    x^2 - x - 3ix - x + 3ix + (1 - 9i^2) =

    x^2 - 2x + 10


    ... and, yes, I performed long division.
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  5. #5
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    Re: Need some help with finding zeros of a polynomial given one complex zero

    Thank you! This makes so much sense!
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