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Octic Sum Settlement (Posted on 2015-02-06) Difficulty: 3 of 5
The roots of the equation x3 - x + 1 = 0 are P, Q, R.

Find P8 + Q8 + R8.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Solution Comment 3 of 3 |
Let S(n) denote the sum of the nth powers of the roots.  Then the polynomial implies S(n) = S(n-2)-S(n-3).  

S(0)=3 is trivial.
S(1)=0 from the coefficient of x^2 being 0.
Let C be the cross-product of the roots (C=P*Q+P*R+Q*R), which makes C=-1 from the coefficient of x.  Then S(2) = S(1)^2-2*C = 0^2 - 2*(-1) = 2.

The sequence S(n) starting at n=0 is 3, 0, 2, -3, 2, -5, 5, -7, 10, ...
S(8) = 10, the answer to the problem.

  Posted by Brian Smith on 2017-06-24 09:27:16
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