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Find-a-R-Such-that-x-1-16-x-2-16-x-3-16-30-Where-x-1-x-2-x-3-are-the-roots-of-the-equation-x-3-ax-1-0-

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Question Number 178476 by Shrinava last updated on 17/Oct/22
Find a∈R Such that x_1 ^(16) + x_2 ^(16) + x_3 ^(16) = 30 Where x_1 ,x_2 ,x_3 − are the roots of the equation: x^3 + ax + 1 = 0
Finda∈RSuchthatx116+x216+x316=30Wherex1,x2,x3−aretherootsoftheequation:x3+ax+1=0
Answered by Frix last updated on 17/Oct/22
x^3 +ax+1=0 ⇒ x_k =((2(√(−3a)))/3)sin ((2πk+sin^(−1) ((3(√3))/(2(−a)^(3/2) )))/3) with k=1, 2, 3 let α=((3(√3))/(2(−a)^(3/2) )) x_k =((2(√(−3a)))/3)sin ((2πk+sin^(−1) α)/3) x_k ^(16) =((65536a^8 )/(6561))sin^(16) ((2πk+sin^(−1) α)/3) sin ((2π+sin^(−1) α)/3) =(((√3)cos ((sin^(−1) α)/3) −sin ((sin^(−1) α)/3))/2)=(((√3)c−s)/2) sin ((4π+sin^(−1) α)/3) =−(((√3)cos ((sin^(−1) α)/3) +sin ((sin^(−1) α)/3))/2)=−(((√3)c+s)/2) sin ((6π+sin^(−1) α)/3) =sin ((sin^(−1) α)/3) =s ((((√3)c−s)/2))^(16) +(−(((√3)c+s)/2))^(16) +s^(16) = (with c=(√(1−s^2 ))) =((45s^(12) )/2)−((135s^(10) )/2)+((1215s^8 )/(16))−((1701s^6 )/(64))−((5103s^4 )/(512))+((6561s^2 )/(1024))+((6561)/(32768))= (after some work) =((360cos (12×((sin^(−1) α)/3)) −13104cos (6×((sin^(−1) α)/3)) +19305)/(32768))= =((45α^4 )/(512))+((729α^2 )/(1024))+((6561)/(32768))= =((6561)/(32768))−((19683)/(4086a^3 ))+((32805)/(8192a^6 )) multiplied with ((65536a^8 )/(6561)) x_1 ^(16) +x_2 ^(16) +x_3 ^(16) =2a^8 −48a^5 +40a^2 =30 we need to solve a^8 −24a^5 +20a^2 −15=0 I can only approximate a_1 ≈−.718638134 a_2 ≈2.85282288 no other real solutions
x3+ax+1=0⇒xk=2−3a3sin2πk+sin−1332(−a)3/23withk=1,2,3letα=332(−a)3/2xk=2−3a3sin2πk+sin−1α3xk16=65536a86561sin162πk+sin−1α3sin2π+sin−1α3=3cossin−1α3−sinsin−1α32=3c−s2sin4π+sin−1α3=−3cossin−1α3+sinsin−1α32=−3c+s2sin6π+sin−1α3=sinsin−1α3=s(3c−s2)16+(−3c+s2)16+s16=(withc=1−s2)=45s122−135s102+1215s816−1701s664−5103s4512+6561s21024+656132768=(aftersomework)=360cos(12×sin−1α3)−13104cos(6×sin−1α3)+1930532768==45α4512+729α21024+656132768==656132768−196834086a3+328058192a6multipliedwith65536a86561x116+x216+x316=2a8−48a5+40a2=30weneedtosolvea8−24a5+20a2−15=0Icanonlyapproximatea1≈−.718638134a2≈2.85282288nootherrealsolutions
Commented by Shrinava last updated on 17/Oct/22
Sorry professor, x_1 ^(16) +x_2 ^(16) +x_3 ^(16) =90
Sorryprofessor,x116+x216+x316=90
Commented by Ar Brandon last updated on 17/Oct/22
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Commented by Frix last updated on 17/Oct/22
x_1 ^(16) +x_2 ^(16) +x_3 ^(16) =2a^8 −48a^5 +40a^2 =90 a^8 −24a^5 +20a^2 −45=0 a_1 =−1 a_2 ≈2.85944202 no other real solutions
x116+x216+x316=2a8−48a5+40a2=90a8−24a5+20a2−45=0a1=−1a2≈2.85944202nootherrealsolutions

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