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Math Help - factoring 3rd order polynomial...

  1. #1
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    Unhappy factoring 3rd order polynomial...

    hey can someone remind me how you get the factors of something like this:

    -x^3 + 5x^2 - 5x - 24 = 0

    thank you for your help, its been so long since idone these i tried googling but could not find a systematic approach,
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  2. #2
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    Quote Originally Posted by jonbob81 View Post
    hey can someone remind me how you get the factors of something like this:

    -x^3 + 5x^2 - 5x - 24 = 0

    thank you for your help, its been so long since idone these i tried googling but could not find a systematic approach,
    As far as I can see -x^3 + 5x^2 - 5x - 24 has no 'simple' factors. A general approach would be to use the cubic formula, but often that's like killing a fly with an elephant gun. Sometimes grouping will work, sometimes simple linear factors can be found by inspection.
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  3. #3
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    i was thinking the same. i was trying to produce the eigenvalues of this matrix

    call lambda x

    (2-x ) 0 3
    0 (6-x) -3
    3 -3 (2-x)

    and that formula below was what i got
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  4. #4
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    Quote Originally Posted by jonbob81 View Post
    hey can someone remind me how you get the factors of something like this:

    -x^3 + 5x^2 - 5x - 24 = 0

    thank you for your help, its been so long since idone these i tried googling but could not find a systematic approach,
    If this has rational roots (and so nice factors) by the rational root theorem the roots are amoung the factors of 24, so try +/11, +/-2, +/-3, +/-4, +/-6, +/-8, +/-12. If none of these are roots then it has no rational roots and so no linear factors with integer coefficients.

    (As it happens none of these are roots and so this cubic has no nice factors, but it does have exactly one real root)

    You can find the root algebraicaly by using the Cardano-Tartaglia cubic formula, though you will probably be better off solving for the real root numerically

    CB
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  5. #5
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    Quote Originally Posted by jonbob81 View Post
    i was thinking the same. i was trying to produce the eigenvalues of this matrix

    call lambda x

    (2-x ) 0 3
    0 (6-x) -3
    3 -3 (2-x)

    and that formula below was what i got
    Quote Originally Posted by jonbob81 View Post
    [snip]
    -x^3 + 5x^2 - 5x - 24 = 0

    [snip]
    Your formula is wrong. The determinant is (2 - x)[(6 - x)(2 - x) - 9] + 3[-3(6 - x)] = -x^3 + 10x^2 - 10x - 48, which factorises as -(x - 8)(x^2 - 2x - 6).

    Solving -(x - 8)(x^2 - 2x - 6) = 0 is simple.
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    Your formula is wrong. The determinant is (2 - x)[(6 - x)(2 - x) - 9] + 3[-3(6 - x)] = -x^3 + 10x^2 - 10x - 48, which factorises as -(x - 8)(x^2 - 2x - 6).

    Solving -(x - 8)(x^2 - 2x - 6) = 0 is simple.


    hey thanks for that, i realised that later and forgot to change it ...silly me
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