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Math Help - verify?!

  1. #1
    Member i_zz_y_ill's Avatar
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    verify?!

    I cannot verify this formula. Can't figure it out at all even though its seems simple Q
    z=\sqrt[3]{(4+\sqrt15)} +\sqrt[3]{(4-\sqrt15)} satisfies

    {z}^3-3z-8=0
    Last edited by i_zz_y_ill; October 2nd 2008 at 05:34 AM. Reason: letx
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  2. #2
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    Hello, i_zz_y_ill!

    Show that: . z\:=\:\sqrt[3]{(4+\sqrt15)} +\sqrt[3]{(4-\sqrt15)} .satisfies . z^3-3z-8\:=\:0

    Let: . \begin{array}{ccccccc} u \;=\; \sqrt[3]{4 + \sqrt{15}} & \Rightarrow & u^3 \;=\; 4 + \sqrt{15} \\ \\[-4mm]<br />
v \;=\; \sqrt[3]{4-\sqrt{15}} & \Rightarrow & v^3 \;=\; 4 - \sqrt{15} \end{array} . . . Hence: . z \:=\:u+v


    Then: . u^3 + v^3 \:=\:8

    . and: . uv \:=\:\sqrt[3]{(4 +\sqrt{15})(4-\sqrt{15})}  \;=\;\sqrt[3]{16-15} \;=\;\sqrt[3]{1} \;=\;1


    We have: . z^3 - 3z - 8 \;=\;(u+v)^3 - 3(u+v) - 8

    Expand: . u^3 + 3u^2v + 3uv^2 - 3(u+v) -8

    \text{Re-arrange terms: }\;\underbrace{(u^3+v^3)}_{\text{This is 8}} \;+ \;3\underbrace{(uv)}_{\text{This is 1}}(u+v) \;- \;3(u+v) \;-\;8

    . . . . . . . . . . =\;\;8 + 3(u+v) - 3(u+v) -8\;\;=\;\;\boxed{0} \quad\hdots\;\text{There!}

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