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Summing powers (Posted on 2004-07-01) Difficulty: 3 of 5
I thought of three numbers.
Their sum is 6.
The sum of their squares is 8.
The sum of their cubes is 5.
What is the sum of their fourth powers?

  Submitted by Federico Kereki    
Rating: 3.0000 (5 votes)
Solution: (Hide)
Solution from http://www.qbyte.org/puzzles/p079s.html

Let the numbers be a, b, and c. Then we have

a+b+c=6
a²+b²+c²=8
a³+b³+c³=5

We will find the monic cubic equation whose roots are a, b, and c. If cubic equation x³-Ax²+Bx-C=0 has roots a, b, c, then, expanding (x-a)(x-b)(x-c), we find

A = a + b + c
B = ab + bc + ca
C = abc

Then B=ab+bc+ca=½[(a+b+c)²-(a²+b²+c²)]=14. Hence a, b, c are roots of x³-6x²+14x-C=0, and we have

a³-6a²+14a-C=0
b³-6b²+14b-C=0
c³-6c²+14c-C=0

Adding, we have (a³+b³+c³) -6(a²+b²+c²) +14(a+b+c) -3C =5-6×8+14×6-3C =0. Hence C=41/3, and x³ -6x² +14x-41/3 =0.

Multiplying the polynomial by x, we have x^4-6x³ +14x² -41x/3 =0. Then

a^4-6a³+14a²-41a/3=0
b^4-6b³+14b²-41b/3=0
c^4-6c³+14c²-41c/3=0

Adding, we have (a^4+b^4+c^4) -6(a³+b³+c³) +14(a²+b²+c²) -41(a+b+c)/3 =0.

Hence a^4+b^4+c^4 =6×5-14×8+(41/3)×6=0.

That is, the sum of the fourth powers of the numbers is 0.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionComplete analytical solutionDanish Ahmed Khan2012-10-24 14:33:51
SolutionSolution To Sum Of Sixth PowersK Sengupta2007-05-31 05:47:12
SolutionAn Alternative MethodologyK Sengupta2007-05-31 05:44:21
AnswerK Sengupta2007-05-31 05:43:01
Questiona b & cJim2004-07-15 16:45:21
Hints/Tipsre: Follow up questionJim2004-07-15 16:42:09
Some Thoughtscomplicated fractions?Jim2004-07-15 16:18:01
Solutionsolution (the easy way)Jim2004-07-15 15:55:57
No SubjectLeigh Lillico2004-07-09 18:59:13
Some Thoughtsre: ImagineNick Hobson2004-07-06 19:13:49
ImagineLarry2004-07-05 02:21:13
John LennonLarry2004-07-04 00:54:23
QuestionFollow up questionNick Hobson2004-07-03 16:40:08
re: Solution (calculation error)np_rt2004-07-01 15:11:10
Solutionnp_rt2004-07-01 13:35:33
Some ThoughtsSolution (?)Eric2004-07-01 10:22:52
re: question for KerekiRenee2004-07-01 09:40:23
question for KerekiRenee2004-07-01 09:29:46
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