document.write( "Question 1209722: Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
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Algebra.Com's Answer #850029 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
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\n" ); document.write( "\n" ); document.write( "There is another method to solve, so beautiful that
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document.write( "        Notice that since the constant term is not zero, \r\n" );
document.write( "           no one root p, q, r  or  s  is zero.\r\n" );
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document.write( "Reduce the given equation to the standard form combining like terms.  You will get\r\n" );
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document.write( "    g(x) = x^4 + 2x^3 + 16x^2 + 20x - 31.    (1)\r\n" );
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document.write( "Divide this reduced equation by x^4.  You will get another polynomial (like a sock turned inside out)\r\n" );
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document.write( "     +  +  +  - .    (2)\r\n" );
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document.write( "If some value p, q, r, s  is the root to polynomial (1),\r\n" );
document.write( "then 1/p, 1/q, 1/r, 1/s  is the zero of function (2).\r\n" );
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document.write( "Let's consider now the polynomial\r\n" );
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document.write( "    h(y) = 1 + 2y + 16y^2 + 20y^3 - 31y^4.    (3)\r\n" );
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document.write( "Compare (3) with (2) and recognize that (3) is the same expression as (2) with replaced  '1/x' by  'y'.\r\n" );
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document.write( "Since p, q, r and s are the roots for polynomial (1),\r\n" );
document.write( "1/p, 1/q, 1/r and 1/s are the roots for polynomial (3).\r\n" );
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document.write( "Now apply Vieta's theorem and find that the sum of the roots 1/p, 1/q, 1/r, 1/s\r\n" );
document.write( "is equal to the coefficient 20 at y^3 in polynomial (3), divided by the leading coefficient -31 at y^4,\r\n" );
document.write( "taken with the opposite sign\r\n" );
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document.write( "    1/p + 1/q + 1/r + 1/s =  = .\r\n" );
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document.write( "At this point, the problem is solved in full, without making cumbersome calculations.\r\n" );
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document.write( "ANSWER.  1/p + 1/q + 1/r + 1/s = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Isn't it beautiful ?\r
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\n" ); document.write( "\n" ); document.write( "This method is called \"turning a polynomial inside out\".\r
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\n" ); document.write( "\n" ); document.write( "Turning a polynomial inside out can be done mentally,
\n" ); document.write( "so one can write an answer immediately, without making/writing these reasons on paper.\r
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\n" ); document.write( "\n" ); document.write( "If you show this focus-pocus to your teacher/professor or at the interview,
\n" ); document.write( "the other side will be shocked to see such an elegant solution.\r
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