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Roots Ready to Relate (Posted on 2016-11-06) Difficulty: 3 of 5

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

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

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Solution Solution Comment 1 of 1
Denote the various sums of products of roots as follows:

S1 = a + b + c + d

S2 = ab + ac + ad + bc + bd + cd

S3 = abc + abd + acd + bcd

S4 = abcd

Expanding S33 gives (after much algebra):

S33 = (abc)3 + (abd)3 + (acd)3 + (bcd)3 + 3S4(S2S3 – S1S4)

By using the coefficients of the quartic,

S1 = 15/24,   S2 = 0,   S3 = 0,   S4 = 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 2016-11-06 10:51:26
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