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Math Help - Solution to an equation with high exponents

  1. #1
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    Solution to an equation with high exponents

    let w be a solution of the equation x^2 + x + 1=0. Then what is w^10 + w^5 + 3 =?

    I don't even know where to start except to use the quadratic formula. to solve for a w. And more so, my w already involves irrational numbers.... (sqrt(-3))
    And I heard from a source that the only way to solve this was to convert it to polar form? Then use some theorems? Is that the only way to do it?
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  2. #2
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    Quote Originally Posted by gundanium View Post
    let w be a solution of the equation x^2 + x + 1=0. Then what is w^10 + w^5 + 3 =?

    I don't even know where to start except to use the quadratic formula. to solve for a w. And more so, my w already involves irrational numbers.... (sqrt(-3))
    And I heard from a source that the only way to solve this was to convert it to polar form? Then use some theorems? Is that the only way to do it?
    Hi gundanium,

    Why can't you just solve x^2+x+1=0 for its two complex roots and substitute them into w^{10}+w^5+3 and see what happens.

    Or am I missing something here!?

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    Quote Originally Posted by masters View Post
    Hi gundanium,

    Why can't you just solve x^2+x+1=0 for its two complex roots and substitute them into w^{10}+w^5+3 and see what happens.

    Or am I missing something here!?

    Not really, it's just that it seems long, studious and impractical for expanding a root to its 10th power for a 5 point question out of a 100points... So I just thought there would be an easier way.
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  4. #4
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    Quote Originally Posted by gundanium View Post
    let w be a solution of the equation x^2 + x + 1=0. Then what is w^10 + w^5 + 3 =?

    I don't even know where to start except to use the quadratic formula. to solve for a w. And more so, my w already involves irrational numbers.... (sqrt(-3))
    And I heard from a source that the only way to solve this was to convert it to polar form? Then use some theorems? Is that the only way to do it?
    1. Split the term T(w):

    T(w)=w^{10} + w^5 + 3 = (w^{10}+w^5+1)+2

    2. You can prove by substitution that

    w^{10}+w^5+1 = 0~\text{iff}~x^2+x+1=0

    3. Therefore the value of T(w) under the given condition is 2.
    Last edited by CaptainBlack; May 14th 2011 at 12:35 AM.
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  5. #5
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    Quote Originally Posted by earboth View Post
    1. Split the term T(w):

    T(w)=w^{10} + w^5 + 3 = (w^{10}+w^5+1)+2

    2. You can prove by substitution that

    w^{10}+w^5+1 = 0~\text{iff}~x^2+x+1=0

    3. Therefore the value of T(w) under the given condition is 2.
    "You can prove by substitution?"
    If this is a proper solution this is what I need
    But I just don't understand it yet.
    What/How do you mean that
    w^{10}+w^5+1 = 0
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  6. #6
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    Quote Originally Posted by gundanium View Post
    "You can prove by substitution?"
    If this is a proper solution this is what I need
    But I just don't understand it yet.
    What/How do you mean that
    w^{10}+w^5+1 = 0
    use the substitution ...

    x = w^5
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