document.write( "Question 1172300: Given that x^2 - 11x + 28 is a factor of x^4 + k(x^3) - 67(x^2) + 394x - 504, evaluate the sum of the four roots of the equation x^4 + k(x^3) - 67(x^2) + 394x - 504 = 0 \n" ); document.write( "

Algebra.Com's Answer #797289 by ikleyn(52798) $\"\"$ $\"About$

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document.write( "(1)  x^2 - 11x + 28 = factoring = (x-7)*(x-4).    (1)\r\n" );
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document.write( "(2)  You are given that \r\n" );
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document.write( "         x^2 - 11x + 28 is a factor of x^4 + k(x^3) - 67(x^2) + 394x - 504.\r\n" );
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document.write( "     It means that each factor of  (1),  (x-7) and (x-4)  are factors of the polynomial  \r\n" );
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document.write( "         p(x) = x^4 + k(x^3) - 67(x^2) + 394x - 504.      (2) \r\n" );
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document.write( "(3)  In turn, due to the remainder theorem, it means that  the value of x= 4 (as well as the value of x= 7) \r\n" );
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document.write( "     is the root of the polynomial (2).  It gives you an equation for k\r\n" );
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document.write( "         p(4) = 0 = 4^4 + k*4^3 - 67*4^2 + 394*4 - 504.\r\n" );
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document.write( "     It is the same as\r\n" );
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document.write( "                256 + 64k = 0,\r\n" );
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document.write( "     which implies  k = -256/64 = -4.\r\n" );
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document.write( "     So,  k = -4.\r\n" );
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document.write( "(4)  The sum of the roots of the equation \r\n" );
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document.write( "         x^4 + k(x^3) - 67(x^2) + 394x - 504 = 0      (3)\r\n" );
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document.write( "     is equal to the coefficient value at x^3  with the opposite sign  (the Vieta's theorem).\r\n" );
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document.write( "     It means that the sum of the roots of the equation (3) is equal to -k = 4.\r\n" );
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document.write( "ANSWER.  The sum of the roots of the equation  x^4 + k(x^3) - 67(x^2) + 394x - 504 = 0  is equal to 4.\r\n" );
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\n" ); document.write( "\n" ); document.write( " * * * The miracle is that I answered the problem's question WITHOUT solving equation (3). * (!) * (!) * \r
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