1. Sep 5, 2005

mlowery

My teacher assigned us this problem (it is for extra credit, but I'll take what I can get):
$$P = a(b^4-c^4)+b(a^4-c^4)+c(a^4-b^4)$$

My work:
1. $$P = a( (b^2)^2 - (c^2)^2 )+b( (a^2)^2 - (c^2)^2 ) + c( (a^2)^2 - (b^2)^2 )$$

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

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$$P = a[ (b^2+c^2) (b+c) (b-c) ] + b[ (a^2+c^2) (a+c) (a-c) ] + c[ (a^2+b^2) (a+b) (a-b) ]$$

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

Thanks,
Mitch

2. Sep 9, 2005

CarlB

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:

$$P = a(b^4-c^4)+b(c^4-a^4)+c(a^4-b^4)$$

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.

Carl

3. Sep 9, 2005

Gokul43201

Staff Emeritus
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) ?