document.write( "Question 1124747: The roots of the polynomial equation 2x^3 - 8x^2 + 3x + 5 = 0 are alpha, beta and gamma.
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Algebra.Com's Answer #741078 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "The given equation\r\n" );
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document.write( "     = 0        (1)\r\n" );
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document.write( "is equivalent to\r\n" );
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document.write( "     = 0     (2)  (all the coefficients of (1) are divided by 2)\r\n" );
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document.write( "Equation (2) has the same roots  ,    and    as equation (1).  Therefore, \r\n" );
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document.write( "     = ,                       (3)\r\n" );
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document.write( "and, according to Vieta's theorem\r\n" );
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document.write( "     = 4,   = 1.5,   = -2.5.      (4)\r\n" );
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document.write( "Now, an equation with the roots  ,    and    is\r\n" );
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document.write( "     = 0.                     (5)\r\n" );
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document.write( "By the Vieta's theorem (or by applying FOIL directly), the coefficients of the left side polynomial are\r\n" );
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document.write( "      at  x^2;                          (6)\r\n" );
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document.write( "      at x;   and                 (7)\r\n" );
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document.write( "      as the constant term.                (8)\r\n" );
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document.write( "So, my task now is to express the coefficient (6), (7) and (8)  via  the coefficients (4) of the equation (2).\r\n" );
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document.write( "Regarding   ,  it is easy:\r\n" );
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document.write( "     =  =  = 16-3 = 13.\r\n" );
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document.write( "So, the coefficient at x^2 of the polynomial (5)  is   = -13.\r\n" );
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document.write( "Regarding  ,  it is easy, too :\r\n" );
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document.write( "     =  =  = 6.25.\r\n" );
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document.write( "So, the constant term of the polynomial (5)  is   = -6.25.\r\n" );
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document.write( "Regarding  , it is slightly more long way :\r\n" );
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document.write( "     = 1.5  of (4)  implies (squaring both sides)\r\n" );
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document.write( "    2.25 =  = \r\n" );
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document.write( "         =  +  = substituting the known values from (4) = \r\n" );
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document.write( "         =  + 2*(-2.5)*4,\r\n" );
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document.write( "which implies\r\n" );
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document.write( "     = 2.25 + 20 = 22.25.\r\n" );
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document.write( "Thus we know all three coefficients of the polynomial (5)\r\n" );
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document.write( "     = -13  at  x^2;          \r\n" );
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document.write( "     = 22.25 at x;   and \r\n" );
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document.write( "     = -6.25 as the constant term.   \r\n" );
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document.write( "Answer.  The polynomial equation under the question is   = 0.\r\n" );
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