document.write( "Question 1178729: Let r, s, and t be the roots of the equation x^3 - 2x + 1 = 0 in some order. What is the maximal value of r^3 - s- t? \n" ); document.write( "

Algebra.Com's Answer #808108 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Let r, s, and t be the roots of the equation x^3 - 2x + 1 = 0 in some order.
\n" ); document.write( "What is the maximal value of r^3 - s- t?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " The solution in the post by @MathLover1 is not sufficient.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I came to bring the correct solution.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "First, you need to read, to interpret and to understand the condition correctly.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    +------------------------------------------------------------------------------------+\r\n" );
document.write( "    |    The problems asks to find the maximal value of the expression of  r^3 - s - t   |\r\n" );
document.write( "    |                                                                                    |\r\n" );
document.write( "    |    over ALL POSSIBLE PERMUTATIONS of the roots r, s and t.                         |\r\n" );
document.write( "    +------------------------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To find the value of  r^3 - s - t  for only one possible permutation, as @MathLover1 does,\r\n" );
document.write( "IS NOT ENOUGHT to solve the problem.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, if r is one of the roots, then  r^3 - 2r + 1 = 0,  which implies\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    r^3 = 2r - 1  and further  r^3 - s - t = (2r-1) - s - t = 3r - (r + s + t) - 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The sum of the roots  (r + s + t) is the coefficient at x^2 of the original equation, taken with\r\n" );
document.write( "the opposite sign (the Vieta's theorem).\r\n" );
document.write( "\r\n" );
document.write( "In our case, this coefficient is zero;  therefore\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    r^3 - s - t = 3r - (r + s + t) - 1 = 3r - 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "THEREFORE,  the expression  r^3 - s - t  is maximal when 3r  is maximal, or, equivalently,  \r\n" );
document.write( "when the root  \"r\"  is maximal of the three roots.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The roots of the equation  x^3 - 2x + 1 = 0  are  1,    and  ,\r\n" );
document.write( "\r\n" );
document.write( "    as @MathLover1 did find in her post.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Of them, the root  1  has maximum value.\r\n" );
document.write( "\r\n" );
document.write( "THEREFORE, from what is written above in my post, the maximal value of the expression  r^3 - s - t  is  3*1 - 1 = 2.   \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  Under given conditions, the maximal value of the expression  r^3 - s - t  is  2.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved (correctly).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );