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Math Help - roots of polynomial equations (complex #s)

  1. #1
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    roots of polynomial equations (complex #s)

    so im just starting to learn about roots of polynomial equations with roots that are complex numbers and i dont understand them at all.. i hope you can help

    A. Use the quadratic formula to find the roots of the equation x^2+4x+5=0. Simplify and compare the roots. what do you notice?
    B. write a quadratic equation with integral coefficients such that one of its roots is 4-5i
    C. Write a quartic equation with integral coefficients and with roots 7i and -3i
    .....
    so for A i think its x= -2+i and x=-2-i but i dont "notice" anything and have no idea how to do the other questions.
    im thinking its something simple and im going to feel stupid once i find out the answer -.-
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  2. #2
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    The roots are conjugate pairs
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  3. #3
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    Quote Originally Posted by phthiriasis View Post
    so im just starting to learn about roots of polynomial equations with roots that are complex numbers and i dont understand them at all.. i hope you can help

    A. Use the quadratic formula to find the roots of the equation x^2+4x+5=0. Simplify and compare the roots. what do you notice?
    B. write a quadratic equation with integral coefficients such that one of its roots is 4-5i
    C. Write a quartic equation with integral coefficients and with roots 7i and -3i
    .....
    so for A i get x= -2+i and x=-2-i but i dont "notice" anything and have no idea how to do the other questions.
    im thinking its something simple and im going to feel stupid once i find out the answer -.-

    What they wanted you to notice is that -2+i \mbox{ and } -2-i

    are conjugates two complex numbers are conjugate if

    a+bi \mbox{ and } a-bi the numbers are the same, but the immaginary parts have opposite signs.

    Now with this new knowlege

    we can find the roots of the other quadratic.

    since we know that 4-5i is a root then 4+5i must also be a root.

    If c is a root of a polynomial then x-c is a facor of the polynomial

    now comes the fun part

    We multiply out our two factors.

    [x-(4-5i)][x-(4+5i)]=x^2-(4+5i)x-(4-5i)x+(4-5i)(4+5i)=
    x^2-4x-5ix-4x+5ix+16+20i-20i-25i^2=x^2-8x+16-25(-1)=
    x^2-8x+41

    I hope this helps
    see what you can do with the next one.
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  4. #4
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    thanks a lot TheEmptySet
    so i tried the next one..
    since 7i and -3i are roots the other roots are -7i and 3i
    so i multiplied it all out..
    (x-7i)(x+7i)(x-3i)(x+3i) and ended up with
    x^4+58x^2+441
    is this right?^^
    thanks again!
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