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Thread: [SOLVED] Roots in Polar Form

  1. #1
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    [SOLVED] Roots in Polar Form

    Hi

    If I am calculating roots in polar form how should I be displaying them?

    eg. The seventh roots of -128 are:

    $\displaystyle z_{0} = \langle2,\pi/7\rangle$
    $\displaystyle z_{1} = \langle2,3\pi/7\rangle$
    $\displaystyle z_{2} = \langle2,5\pi/7\rangle$
    $\displaystyle z_{3} = \langle2,\pi\rangle = -2$
    $\displaystyle z_{4} = \langle2,9\pi/7\rangle$
    $\displaystyle z_{5} = \langle2,11\pi/7\rangle$
    $\displaystyle z_{6} = \langle2,13\pi/7\rangle$

    Do I need to be displaying them like this?

    $\displaystyle z_{0} = \langle2,\pi/7\rangle = 2(cos(\pi/7)+isin(\pi/7))$

    etc, etc...

    Or like in the first example

    The reason I ask is because i'm asked to show how they make 3 complex conjugate pairs which I can see are

    $\displaystyle z_{0}$ and $\displaystyle z_{6}$
    $\displaystyle z_{1}$ and $\displaystyle z_{5}$
    $\displaystyle z_{2}$ and $\displaystyle z_{4}$


    Thanks
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  2. #2
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    Quote Originally Posted by Ian1779 View Post
    eg. The seventh roots of -128 are:
    $\displaystyle z_{0} = \langle2,\pi/7\rangle$
    $\displaystyle z_{1} = \langle2,3\pi/7\rangle$
    $\displaystyle z_{2} = \langle2,5\pi/7\rangle$
    $\displaystyle z_{3} = \langle2,\pi\rangle = -2$
    $\displaystyle z_{4} = \langle2,9\pi/7\rangle$
    $\displaystyle z_{5} = \langle2,11\pi/7\rangle$
    $\displaystyle z_{6} = \langle2,13\pi/7\rangle$

    Do I need to be displaying them like this?

    $\displaystyle z_{0} = \langle2,\pi/7\rangle = 2(cos(\pi/7)+isin(\pi/7))$
    I think that really up to your instructor/textbook

    The reason I ask is because i'm asked to show how they make 3 complex conjugate pairs which I can see are
    $\displaystyle z_{0}$ and $\displaystyle z_{6}$
    $\displaystyle z_{1}$ and $\displaystyle z_{5}$
    $\displaystyle z_{2}$ and $\displaystyle z_{4}$
    As to the complex conjugate pairs, take note.
    $\displaystyle \arg (z) = - \arg \left( {\overline z } \right)$
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  3. #3
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    Thanks for the explanation about the arg!!

    Another reason I ask is that the follow on question asks me to factorise the polynomial $\displaystyle x^7+128$

    I've worked out that one of the roots are $\displaystyle 2(cos(\pi/7)+isin(\pi/7))$ and another is $\displaystyle -2$ hence my question about the polar forms as I feel i'm duplicating my work

    Any advice would be great

    Thanks
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  4. #4
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    Quote Originally Posted by Ian1779 View Post
    Another reason I ask is that the follow on question asks me to factorise the polynomial $\displaystyle x^7+128$
    $\displaystyle x^7+128=(x-z_0)(x-z_1)(x-z_2)(x-z_3)(x-z_4)(x-z_5)(x-z_6)$
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  5. #5
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    Thanks
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