Algebra.Com's Answer #851719 by ikleyn(52876)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Consider  .\r
\n" );
document.write( "\n" );
document.write( "By finding the roots in  form, and using appropriate substitutions, show that\r
\n" );
document.write( "\n" );
document.write( " = 0.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "Equation  = 0 is the same as  = i.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One root is, obviously, z = i, since  = i.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let's list all the roots \r\n" );
document.write( "\r\n" );
document.write( "  =  =  , \r\n" );
document.write( "\r\n" );
document.write( "  =  =  =  =  = i, \r\n" );
document.write( "\r\n" );
document.write( " (we just noticed it above !)\r\n" );
document.write( "\r\n" );
document.write( "  =  =  =  , \r\n" );
document.write( "\r\n" );
document.write( "  =  =  =  ,\r\n" );
document.write( "\r\n" );
document.write( "  =  =  =  .\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Notice that and have opposite real parts and identical imaginary parts. (*)\r\n" );
document.write( "Similarly, and have opposite real parts and identical imaginary parts. (**)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We can write the decomposition of in the form of the product of linear binomials with the roots\r\n" );
document.write( "\r\n" );
document.write( " =  =\r\n" );
document.write( "\r\n" );
document.write( " =  . (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this decomposition (1), second and third parentheses will give the product\r\n" );
document.write( "\r\n" );
document.write( "  =  . (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Here  =  , as we noticed in (*), and  =  =  = -1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, \r\n" );
document.write( "\r\n" );
document.write( " =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Similarly, in decomposition (1), fourth and fifth parentheses will give the product\r\n" );
document.write( "\r\n" );
document.write( "  =  . (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Here  =  , as we noticed in (**), and  =  =  = -1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, \r\n" );
document.write( "\r\n" );
document.write( " =  . (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, combining everything in one piece, we get\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " If = 0, then =  = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "QED.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the proof is complete.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "////////////////////////////\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "In her post, @MathLover1 incorrectly read the problem and incorrectly understood
\n" );
document.write( "what the problem requested to prove.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "So, her writing in her post is not a proof of the problem' statement
\n" );
document.write( "and has nothing in common with what this problem requests to prove.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "For the peace in your mind, simply ignore that post.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |