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Roots Ready to Relate (Posted on 20161106) 

24x^4  15x^3 + 1
Let a,b,c,d be the roots of the polynomial above.
Find the value of (abc)^3 + (abd)^3 + (acd)^3 + (bcd)^3
Solution

Comment 1 of 1

Denote
the various sums of products of roots as follows:
S_{1} = a + b + c + d
S_{2} = ab + ac + ad + bc + bd + cd
S_{3} = abc + abd + acd + bcd
S_{4} = abcd
Expanding S_{3}^{3} gives (after much algebra):
S_{3}^{3} = (abc)^{3} + (abd)^{3} + (acd)^{3}
+ (bcd)^{3} + 3S_{4}(S_{2}S_{3} – S_{1}S_{4})
By using the coefficients of the quartic,
S_{1} = 15/24, S_{2} =
0, S_{3} = 0, S_{4} = 1/24 and we obtain:
0 = (abc)^{3} + (abd)^{3} + (acd)^{3} + (bcd)^{3}
+ 3(1/24)(0 – 5/192)
which gives
(abc)^{3} + (abd)^{3} + (acd)^{3} + (bcd)^{3} =
5/1536 = 0.003255..

Posted by Harry
on 20161106 10:51:26 


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