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Check my answer please (factoring)

  1. Sep 5, 2005 #1
    My teacher assigned us this problem (it is for extra credit, but I'll take what I can get):
    [tex]P = a(b^4-c^4)+b(a^4-c^4)+c(a^4-b^4)[/tex]

    My work:
    1. [tex]P = a( (b^2)^2 - (c^2)^2 )+b( (a^2)^2 - (c^2)^2 ) + c( (a^2)^2 - (b^2)^2 )[/tex]

    2. [tex]P = a[ (b^2+c^2) (b+c) (b-c) ] + b[ (a^2+c^2) (a+c) (a-c) ] + c[ (a^2+b^2) (a+b) (a-b) ][/tex]

    Final Answer
    [tex]P = a[ (b^2+c^2) (b+c) (b-c) ] + b[ (a^2+c^2) (a+c) (a-c) ] + c[ (a^2+b^2) (a+b) (a-b) ][/tex]

    Can someone please double check my work. As far as I can see, that is the simplest form to which it can be factored.

  2. jcsd
  3. Sep 9, 2005 #2


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    Homework Helper

    If I wanted the answer to be "factored", I would think that it should be in the form of a product of sums. Factoring, that is, is the process of turning a sum of products into a product of sums.

    Just looking at the problem, perhaps the teacher meant to write:

    [tex]P = a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)[/tex]

    This is a bit more symmetric, and reminds one of a classic SU(3) symmetry, which should give hints on how to factor it. For example, note that if a=b, then P=0, so (a-b) must be a factor.

    If, on the other hand, the teacher meant what he wrote, then you should consider the effect of making the substitution b => -b'. In other words, (a+b) is a factor for the problem you've written.

  4. Sep 9, 2005 #3


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    I'm almost certain the cyclically symmetric form what what was intended.

    Mitch, do you understand what factoring means ? It means that the given expression be written entirely as a (non-trivial) product of terms - each of these terms called a factor. What you've arrived at is a sum of terms and hence is not what is asked for.

    Look at CarlB's suggestion - do you understand it ? If you have some polynomial (in one variable, say) function, f(x), and if f(u) = 0, then what can one say about (x-u) ?
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