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Factoring (X^4+1)

  1. Jan 22, 2010 #1
    1. The problem statement, all variables and given/known data

    I am trying to factor x4+1 in to two multiplied polynomials

    2. Relevant equations


    My teacher gave us this hint that its factored form is (ax2+bx+c)(ax2+bx+c)


    3. The attempt at a solution

    First i assumed that a and c were equal to 1 so that when x2 is multiplied by the other x2 is gives me x4 and 1 times 1 gives me 1. I knew that b had to be a constant so I multiplied...

    (however i didnt know if both b's were the same so i split them into a and b. I also knew one of the constants must be negative so that variables cancel out.)

    (x2+ax+1)(x2-bx+1)= x4+1

    and I get

    x4+ax3-bx3+2x2-abx2+ax-bx+1= x4+1 canceling terms I get

    ax3-bx3+2x2-abx2+ax-bx=0

    I noticed that to cancel out ax3-bx3 and ax-bx , a and b must be equal to each other. This means 2x2-abx2 = 0

    2=ab(but they are the same) 2=b2

    b=[tex]\sqrt{}2[/tex]

    So i checked my answer and it works out, but I am wondering if there is a more systematic approach to solve this so that I don't have to assume as much as I did.
     
  2. jcsd
  3. Jan 22, 2010 #2

    rock.freak667

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    x4+1 has no real roots, so at most you can factor it into complex roots

    using i2=-1

    x4-i2=(x2)2-(i)2
     
  4. Jan 22, 2010 #3

    tiny-tim

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    Welcome to PF!

    Hi flyers! Welcome to PF! :smile:

    (have a square-root: √ :wink:)
    You could have looked for a way to write it as the difference of two squares …

    so complete the square …

    x4 + 1 = (x2 + 1)2 - 2x2 :wink:
     
  5. Jan 22, 2010 #4

    tiny-tim

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    More generally …

    x4 + 2(a-b)x2 + a2

    = (x2 + a)2 - 2bx2

    = (x2 + (√2b)x + a)(x2 - (√2b)x + a) :smile:
     
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