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| Roots Ready to Relate (Posted on 2016-11-06) |
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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
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Comment 1 of 1
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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..
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Posted by Harry
on 2016-11-06 10:51:26 |
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