Algebra.Com's Answer #849541 by ikleyn(52782)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "If x⁵ = 1 with x ≠ 1, find the value of
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document.write( "1/(1+x²) + 1/(1+x⁴) + 1/(1+x) + 1/(1+x³)
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document.write( "In this problem, all numbers  , for which x^5 = 1, are five different complex roots of equation\r\n" );
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document.write( "  = 1, k = 1, 2, 3, 4, 5. (1)\r\n" );
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document.write( "including real root x = 1. \r\n" );
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document.write( "THREFORE, the numbers  =  are the five complex roots of equation \r\n" );
document.write( "  = 1. (2)\r\n" );
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document.write( "including real number w = 2, which corresponds to real numer 1, which is the root of (1).\r\n" );
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document.write( "Hence, the reciprocal numbers  are the roots of this equation\r\n" );
document.write( "  = 1. (3)\r\n" );
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document.write( "including real number w = 1/2, which corresponds to real numer 1, which is the root of (1).\r\n" );
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document.write( "Equation (3) can be written in equivalent polynomial form\r\n" );
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document.write( "  = w^5. (4)\r\n" );
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document.write( "or\r\n" );
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document.write( " 1 - 5w + 10w^2 - 10w^3 + 5w^4 - w^5 = w^5,\r\n" );
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document.write( "or\r\n" );
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document.write( " 2w^5 - 5w^4 + 10w^3 - 10w^2 + 5w - 1 = 0. (5)\r\n" );
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document.write( "So, the sum of four addends\r\n" );
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document.write( "  +  +  +  (6)\r\n" );
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document.write( "is the sum of all four complex roots of equation (5), that are not real numbers.\r\n" );
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document.write( "Let's add  to the sum (6). We will get the sum S\r\n" );
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document.write( " S = + + + + . (7)\r\n" );
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document.write( "Now this sum (7) is the sum of all complex roots of equation (5).\r\n" );
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document.write( "According to Vieta's theorem, this sum (7) is the coefficient at  divided by the coefficient at x^5 \r\n" );
document.write( "of equation (5), taken with the opposite sign.\r\n" );
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document.write( "This ratio of the mentioned coefficients in (5) is  =  .\r\n" );
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document.write( "Hence, the sum (7) is equal to - =  .\r\n" );
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document.write( "But the problem asks about S - , and we finally find this value \r\n" );
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document.write( " S - = + + + = - =  = 2.\r\n" );
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document.write( "At this point, the problem is solved completely, to the very end.\r\n" );
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document.write( "ANSWER. If x⁵ = 1 with x ≠ 1, then the sum 1/(1+x²) + 1/(1+x⁴) + 1/(1+x) + 1/(1+x³) is 2.\r\n" );
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document.write( "Solved.\r
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document.write( "This problem/solution is of the MOST UPPER LEVEL of the problems of this kind.\r
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document.write( "By the way, my solution shows that the problem can be generalized
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document.write( "for more high degrees in equation (1) and more longer sums
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document.write( "with the similar expressions for the roots.\r
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