How to solve a 4 variable polynomial equation

Asked Oct 14 '19 at 14:20

Active Oct 14 '19 at 14:20

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I was working on a Statistics problem and had been given some data (Mean, variance, Skewness and Kurtosis values). From that, I arrived at the following 4 equations but I am stumped here. Any suggestions would greatly help, thanks !

a + b + c + d = 20
(a-5)² + (b-5)² + (c-5)² + (d-5)² = 46
(a-5)³ + (b-5)³ + (c-5)³ + (d-5)³ = 2105.93
(a-5)⁴ + (b-5)⁴ + (c-5)⁴ + (d-5)⁴ = 118760.5

asked Oct 14 '19 at 14:20

Isildur X1

Start with $$a-5=A$$ etc. – lab bhattacharjee Oct 14 '19 at 14:24
From where does it come? – Dr. Sonnhard Graubner Oct 14 '19 at 14:24
I would use the Goebner Method – Dr. Sonnhard Graubner Oct 14 '19 at 14:25
1

Step 1: Calculate $a^i+b^i+c^i+d^i$ for $i=1,2,3,4$.
Step 2: Calculate the symmetric functions in $a,b,c,d$.

Step 3: Construct & solve the quartic equation. ... Good luck $ \ddot \smile$
– Donald Splutterwit Oct 14 '19 at 14:26
I think it is impossible to solve it by hand. – Dr. Sonnhard Graubner Oct 14 '19 at 14:47
System reduced to $-15519445 - 583186 x + 76200 x^2 - 12000 x^3 + 600 x^4=0$, where $x=(a,b,c,d)$. – Dmitry Ezhov Oct 14 '19 at 14:52
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@Dr.SonnhardGraubner You should stop using WolframAlpha so much and read about symmetric polynomials and Ferrari's formulas. – conditionalMethod Oct 14 '19 at 15:03
I know all these formulas,but i think they will not help at all. – Dr. Sonnhard Graubner Oct 14 '19 at 15:38

How to solve a 4 variable polynomial equation

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