perplexus.info :: Just Math : Octic Sum Settlement

Octic Sum Settlement (Posted on 2015-02-06)

The roots of the equation x³ - x + 1 = 0 are P, Q, R.

Find P⁸ + Q⁸ + R⁸.

See The Solution Submitted by K Sengupta

Rating: 4.5000 (2 votes)

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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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