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Math Help - Factorise Polynomial

  1. #1
    Member courteous's Avatar
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    Factorise Polynomial

    x^4-x^2+1 Factorize.
    I've set x^2=t \rightarrow t^2-t+1=0. Using the formula, I get x_{1,2}=\pm\sqrt\frac{1+i\sqrt3}{2}  x_{3,4}=\pm\sqrt\frac{1-i\sqrt3}{2} What now?!
    Last edited by courteous; September 14th 2008 at 05:26 AM. Reason: TYPO: plus -> minus
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  2. #2
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by courteous View Post
    x^4-x^2+1 Factorize.
    I've set x^2=t \rightarrow t^2-t+1=0. Using the formula, I get x_{1,2}=\pm\sqrt\frac{1+i\sqrt3}{2}  x_{3,4}=\pm\sqrt\frac{1-i\sqrt3}{2} What now?!
    x^4-x^2+1=(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})

    is what you want
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  3. #3
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    \pm\sqrt\frac{1+i\sqrt3}{2} can be rewritten as \pm\sqrt{e^{i\frac{\pi}{3}}} and then you can determine the roots of that to be e^{i\frac{\pi}{6}} and e^{i\frac{7\pi}{6}}. For the other roots, the approach is similar.
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  4. #4
    Member courteous's Avatar
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    Quote Originally Posted by bkarpuz View Post
    x^4-x^2+1=(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})

    is what you want
    Indeed, but what are x_{1}, x_{2}, x_{3}, x_{4}, and most importantly, how do you get them?

    Quote Originally Posted by icemanfan View Post
    \pm\sqrt\frac{1+i\sqrt3}{2} can be rewritten as \pm\sqrt{e^{i\frac{\pi}{3}}}
    icemanfan, you are beyond me, care to explain? Why is it, that \frac{1+i\sqrt3}{2} = e^{i\frac{\pi}{3}}

    The solution should be (x^2+\sqrt3x+1)(x^2-\sqrt3x+1). How can you get it?
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  5. #5
    Senior Member bkarpuz's Avatar
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    Exclamation

    Quote Originally Posted by courteous View Post
    Indeed, but what are x_{1}, x_{2}, x_{3}, x_{4}, and most importantly, how do you get them?


    icemanfan, you are beyond me, care to explain? Why is it, that \frac{1+i\sqrt3}{2} = e^{i\frac{\pi}{3}}

    The solution should be (x^2+\sqrt3x+1)(x^2-\sqrt3x+1). How can you get it?
    x_{i}(i=1\ldots4) are the roots you have posted in your first message ( see Algebric proofs Section at Fundamental theorem of algebra - Wikipedia, the free encyclopedia )

    x+iy=r\mathrm{e}^{i\theta}, where r=\sqrt{x^{2}+y^{2}} and \theta=\mathrm{atan}(y/x) ( see Euler's formula - Wikipedia, the free encyclopedia ).
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  6. #6
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    what you do is x^4-x^2+1

    and just start doing normal factoring, dont use equations, this way will take you much shorter time...

    unless it states otherwise, this would be the easiest way to do it
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  7. #7
    Member courteous's Avatar
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    Quote Originally Posted by log(xy) View Post
    what you do is x^4-x^2+1

    and just start doing normal factoring, dont use equations, this way will take you much shorter time...
    But what if you do not "see" what the factor is (in this case, (x^2+\sqrt3x+1)(x^2-\sqrt3x+1); some tips to see such beast?), or if introducing "new" variable gets you nowhere (in this case, t^2-t+1=0)?

    What are considered best practices (in this case)?
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