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- Feb 1st 2007, 11:29 AMSorobanQuickie #18

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- Feb 1st 2007, 12:37 PMtopsquark
- Feb 2nd 2007, 01:53 AMearboth
Hello, Soroban,

I'm not quite certain if I remember correctly, but I believe that there exist a "coefficient theorem of Vieta" (that's the name I learned long, long time ago) which says:

If then

Let r be the missing root of the equation and using this theorem I get:

a = -(1 + 2 + 3 + r) = -6 - r which is zero here: thus r = -6

b = 1*2 + 1*3 + 1*r + 2*3 + 2*r + 3*r = 11 + 6r

c = -(1*2*3 + 1*2*r + 2*3*r) = -6 - 8r

d = (-1)^4 * (1*2*3*r) = 6r

According to my results b + d (these are your a+c) = 11+6r + 6r = 11 +12 r = 11 - 72 = -61

But - as I mentioned before - I'm not sure that this attempt is correct.

EB - Feb 2nd 2007, 09:17 AMSoroban
A great algebraic solution, Dan . . . Nice!

EB, Vieta's theorem was used in my book's solution.

. . . Your work was absolutely correct.

Let be the fourth solution.

Vieta says: .

And he says: .

Hence: .

He also says: .

Therefore: .