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Let a, b, c be the (possibly complex) roots of the cubic x^3 - 5x^2 +…
Question
Let a, b, c be the (possibly complex) roots of the cubic x^3 - 5x^2 + 6x - 1 = 0. Compute the exact value of a^5 + b^5 + c^5.
✓ Verified answer: 370checked by our engine — not a guess
Step-by-step solution
By Vieta: e1=a+b+c=5, e2=ab+bc+ca=6, e3=abc=1. Let p_k be the k-th power sum. Newton's identities give:
p1 = e1 = 5.
p2 = e1 p1 - 2 e2 = 25 - 12 = 13.
p3 = e1 p2 - e2 p1 + 3 e3 = 65 - 30 + 3 = 38.
p4 = e1 p3 - e2 p2 + e3 p1 = 190 - 78 + 5 = 117.
p5 = e1 p4 - e2 p3 + e3 p2 = 585 - 228 + 13 = 370.
Final answer370
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