document.write( "Question 1205175: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥
\n" ); document.write( "3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞
\n" ); document.write( "and 𝑟 (≠ 0) are real numbers, express the following in terms of 𝑝, 𝑞 and 𝑟:
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Algebra.Com's Answer #841830 by ikleyn(52937) $\"\"$ $\"About$

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\n" ); document.write( "If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥^3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0,
\n" ); document.write( "where 𝑝, 𝑞 and 𝑟 (≠ 0) are real numbers, express the following in terms of 𝑝, 𝑞 and 𝑟:
\n" ); document.write( "1/𝛼^3 + 1/𝛽^3 + 1/𝛾^3
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\n" ); document.write( "\n" ); document.write( "For simplicity of writing, I will replace $\"alpha\"$ , $\"beta\"$ and $\"gamma\"$ by \"a\", \"b\" and \"c\".\r
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\n" ); document.write( "\n" ); document.write( "So, we are given an equation $\"x%5E3+%2B+px%5E2+%2B+qx+%2B+r\"$ = 0, where p, q and r (=/=0) are real numbers,
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\n" ); document.write( "\n" ); document.write( "They want we find $\"1%2Fa%5E3\"$ + $\"1%2Fb%5E3\"$ + $\"1%2Fc%5E3\"$ .\r
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document.write( "                  Step by step solution\r\n" );
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document.write( "(a)  First, notice that if \"a\" is the solution to polynomial equation  = 0, then\r\n" );
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document.write( "            = 0.   (1)\r\n" );
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document.write( "     Since r =/= 0, the root \"a\" is also not zero, a =/= 0.  In equation (1), divide both sides by .\r\n" );
document.write( "     You will get then\r\n" );
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document.write( "            = 0.\r\n" );
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document.write( "     It means that   is the root of the cubic polynomial equation\r\n" );
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document.write( "            = 0.    (2)\r\n" );
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document.write( "     Similarly,  if \"a\", \"b\" and \"c\" are the roots to equation (1), then  ,   and    are the roots\r\n" );
document.write( "     of equation (2).\r\n" );
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document.write( "(b)  OK.  It means that if \"a\", \"b\" and \"c\"  are the roots of equation  (1),   = 0,\r\n" );
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document.write( "          they want we calculate  , where d, e, and f are the roots of equation (2),   = 0.\r\n" );
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document.write( "(c)  Due to Vieta's theorem, if d, e and f are the roots of equation (2), then\r\n" );
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document.write( "         d + e + f = ,  d*e + d*f + e*f = ,  d*e*f = .    (3)\r\n" );
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document.write( "(d)  For any real numbers d, e, f, the following identity is valid\r\n" );
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document.write( "         =  + 3*(d+e+f)*(de + df + ef) - 3def.    (4)\r\n" );
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document.write( "     It can be checked / proved by direct calculation.\r\n" );
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document.write( "(e)  Now, substitute expressions (3) into (4).  You will get then\r\n" );
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document.write( "         =  +  - .\r\n" );
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document.write( "     It implies    =  +  - ,  or\r\n" );
document.write( "                  \r\n" );
document.write( "                   =  +  - .\r\n" );
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document.write( "(f)  Thus the problem is just solved, and the  ANSWER  is:\r\n" );
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document.write( "     if a, b and c are the roots of equation (1),  then   +  +  =  +  - .\r\n" );
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document.write( "ANSWER.  If a, b and c are the roots of equation  = 0,  \r\n" );
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document.write( "         then   +  +  =  +  - .\r\n" );
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