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Math Help - Roots of a polynomial.

  1. #1
    Super Member Showcase_22's Avatar
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    Roots of a polynomial.

    This is a bizarre question I encountered on a homework sheet:

    Every irreducible polynomial in \Re[x] is of degree 1 or 2. Express x^4+1 as a product of irreducible polynomials in \Re[x]
    The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as "x^2-a" where a is a positive real to bypass this. However, I have no way of working out the roots!
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Showcase_22 View Post
    This is a bizarre question I encountered on a homework sheet:



    The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as "x^2-a" where a is a positive real to bypass this. However, I have no way of working out the roots!
    Actually, the polynomial can be written as a product this way :
    (ax+bx+c)(dx+ex+f)

    since the leading coefficient is 1, you can take a=1 and d=1
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  3. #3
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    Quote Originally Posted by Showcase_22 View Post
    This is a bizarre question I encountered on a homework sheet:



    The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as "x^2-a" where a is a positive real to bypass this. However, I have no way of working out the roots!
    One possible trick is to write X^4+1=(X^4+2X^2+1)-2X^2 = (X^2+1)^2 - (\sqrt{2}X)^2=(X^2+X\sqrt{2}+1)(X^2-X\sqrt{2}+1).
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