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July 27

Factorisation

I have been grappling with factorising this

$x^{4}+22x^{3}+172x^{2}+552x+576=0\,\!$

All four solutions are integers and I have already found, by inspection, that $x+2\,\!$ is one factor.

Dividing through by this leaves me with the cubic

$x^{3}+20x^{2}+132x+288=0\,\!$

though this simplifled version doesn't seem to help much.

I can see that not all solutions to this can be positive; there must be either two positive and two negative - counting repeated roots separately - or they are all negative.

The actual factorisation matters little to me; I would just like to be given hints/ideas on how to go about doing this kind of probem. Algebra man 14:40, 27 July 2007 (UTC)[reply]

We have an article about this very problem. See factor theorem. Richard B 14:51, 27 July 2007 (UTC)[reply]

I was already aware of that; it is how I found

x=-2

to be a solution. I could go on forever checking random numbers by myself; I came here because I thought there may be a more elegant way of solving this problem - I do not view trial and error as elegant. Algebra man 14:55, 27 July 2007 (UTC)[reply]

OK, well there is the cubic formula, but that's not the nicest either. By the way, check your coefficient of x in the cubic equation you have above. I think it should be a fair bit higher. Richard B 15:05, 27 July 2007 (UTC)[reply]

You're right. 132x not 13x. Algebra man 15:08, 27 July 2007 (UTC)[reply]

Here's a hint. Let

x^{4}+22x^{3}+172x^{2}+552x+576=(x+a)(x+b)(x+c)(x+d),\,\!

where a, b, c and d are integers (perhaps negative), but not necessarily distinct. What can you say about the product abcd? Baccyak4H (Yak!) 15:01, 27 July 2007 (UTC)[reply]

I'm sorry to say it but I know this as well. This is the fundamental theorem of algebra and abcd is equal to 576 - I have broken down 576 into product of primes,

2^{6}*3^{2}

, and tried to find combinations of those numbers that multipy to give 576 and add together to give 22. No luck, as yet. Also it only just occurs to me that all the soluitions must be negative. Algebra man 15:07, 27 July 2007 (UTC)[reply]

I think we've been over this already. Either you use the general formula for cubic or quartic equations, or you use the rational root theorem. That is: You know that all roots of the polynomial are factors of 576, so you don't need to find combinations of them - you just need to check every factor (there are 21 negative factors) and see if it is a solution. -- Meni Rosenfeld (talk) 15:32, 27 July 2007 (UTC)[reply]

Of course, this can be made even faster by dividing by x-α whenever you stumble upon a root. -- Meni Rosenfeld (talk) 15:33, 27 July 2007 (UTC)[reply]

(after edit conflict)You can of course cut down the factors to look at. You know that -2 is a solution, so we're just looking at finding 3 factors of 288 that sum to 20. The minimum set of 3 numbers that would sum to 20 is 18,1,1, i.e. no factor can be larger than 18 - and we've just shown that 18 isn't a possible solution - because the other factors would be too low. That cuts it down to 9 possible factors to check. Richard B 15:39, 27 July 2007 (UTC)[reply]

You can also use root-finding algorithms like Newton's method. If you want to do Newton's method, in case you don't know calculus, finding f'(x) for a polynomial can be done by multiplying each element by the power of x (576=576x^0), then decrementing the power. For example, the derivative of x^4+22x^3+172x^2+552x+576 is 4x^3+66x^2+344x+552. — Daniel 15:44, 27 July 2007 (UTC)[reply]

Pardon me for stating the obvious; we are told that all four roots are integers, which strongly suggests this is a posed problem. In that case, the context in which it is posed will be full of ideas for how to factor it, and the point of the exercise is to use those ideas. Instead, we're casting about at random.

Another blindingly obvious fact derives from the observation that all coefficients are positive, including the constant term; therefore it is impossible for a non-negative x to produce zero.

If −1 were a root, then 288−132+20−1 (or 576−552+172−22+1) would equal zero, which is clearly not so. A standard tool for root isolation is the Sturm sequence, which for the reduced polynomial p consists of

{\begin{aligned}&x^{3}+20x^{2}+132x+288,\\&3x^{2}+40x+132,\\&x+6.\end{aligned}}

The negative of a root can be no less than 2 and no greater than 18, so a binary search is very quick. Everything else has already been said by others. --KSmrq^T 21:51, 27 July 2007 (UTC)[reply]

Use the Durand-Kerner method and computer. Bo Jacoby 10:02, 28 July 2007 (UTC).[reply]

Here's how I would do it:

1) Factor 576 into 1x2⁶x3². Since there are no negatives in the polynomial, I assume there are no (x-...) factors.

2) Find posssible factors by multiplying these: (x+1)(x+2)(x+3)(x+4)(x+6)(x+8)(x+9)(x+12)(x+16)(x+18)...

3) Start with the lowest and test each. Remember, after a factor works, it might be in there again, so we need to test for it again in the result. So, I ended up testing for (x+1),(x+2), (x+3), (x+4), and (x+6), with a second test each for (x+2) and (x+6). I get (x+8) as a factor automatically after factoring the rest out. So, I had to test for 7 roots to find 4 roots, not too bad, I'd say. StuRat 22:47, 28 July 2007 (UTC)[reply]