Algebra.Com's Answer #846532 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "If x^2 = x+1 and y^2 = y+1, find x^5 + y^5.
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document.write( " I thought if there is a way to avoid long stupid calculations,\r
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document.write( " and finally got an idea.\r
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document.write( " It seems a robust and elegant and deserves a separate presentation.\r
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document.write( "From given equations, we have (after reducing to standard form quadratic equations)\r\n" );
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document.write( " x^2 - x - 1 = 0,\r\n" );
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document.write( "  =  , (1)\r\n" );
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document.write( "  =  , (2)\r\n" );
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document.write( "  = , (3)\r\n" );
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document.write( "  = . (4)\r\n" );
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document.write( " +---------------------------------------------------------+\r\n" );
document.write( " | The idea is that given equations allow to reduce |\r\n" );
document.write( " | the degrees of x^5 and y^5 |\r\n" );
document.write( " | step by step until we get linear binomial in x and y. |\r\n" );
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document.write( "I will write first step in all details. It is\r\n" );
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document.write( " x^5 = x^2*x^3 = here I replace x^2 by (x+1) and continue = (x+1)*x^3 = x^4 + x^3.\r\n" );
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document.write( "So, we reduced the monomial x^5 to the polynomial x^4 + x^3 of degree 4.\r\n" );
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document.write( "Next steps I will use standard equivalent transformations of polynomials,\r\n" );
document.write( "will extract x^2 everywhere where possible and replace it by (x+1).\r\n" );
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document.write( "I will write my transformations in one long line. In order for do not make breaks\r\n" );
document.write( "to explain every time that \"here I replace x^2 by (x+1) and continue\", I will use\r\n" );
document.write( "special sign of doubled equality \" = = \".\r\n" );
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document.write( " | So, every time as you see this sign \" = = \", read |\r\n" );
document.write( " | it as \"here I replace x^2 by (x+1) and continue\". |\r\n" );
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document.write( "Let's go.\r\n" );
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document.write( " x^5 = x^2*x^3 = = (x+1)*x^3 = x^4 + x^3 = x^2*(x^2+x) = = (x+1)*(x^2+x) = x^3 + x^2 + x^2 + x =\r\n" );
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document.write( " = x*3 + 2x^2 + x = x^2*x + 2x^2 + x = = (x+1)*x + 2*(x+1) + x = x^2 + x + 3x + 2 = = (x+1) + 4x + 2 =\r\n" );
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document.write( " = 5x + 3.\r\n" );
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document.write( "Thus, we have \r\n" );
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document.write( " x^5 = 5x + 3. (5)\r\n" );
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document.write( "Similarly,\r\n" );
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document.write( " y^5 = 5y + 3. (6)\r\n" );
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document.write( "So, x^5 + y^5 = 5x + 5y + 6.\r\n" );
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document.write( "Now substitute values , , , from (1) - (4) into this simple formula and get\r\n" );
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document.write( "  +  =  =  =  ;\r\n" );
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document.write( "  + =  = 5 + 6 = 11;\r\n" );
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document.write( " +  =  = 5 + 6 = 11;\r\n" );
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document.write( " + =  =  =  .\r\n" );
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document.write( "At this point, the problem is solved completely.\r
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document.write( "The given equations allow to start (to turn on) the process of reducing degrees
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document.write( "in the formulas, which quickly leads to the desired result.\r
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document.write( "It makes the solution straightforward, easy and elegant.\r
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document.write( "I hope that after this my solution you will see the problem from a completely
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document.write( "different perspective and from a completely different angle of view.\r
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