document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177626>Question 1177626</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The product of three integers is 90. The second number is twice the first number. The third number is two more than the first number. What are the three numbers? Use a polynomial and the calculator to answer the question.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #806765 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52809)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=806765\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "When you guess the solution, then, even if you guessed it correctly, &nbsp;the question remains if the guessed solution is unique.\r<BR>\n" );
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document.write( "Without answering this remaining question, &nbsp;the solution can not be considered as perfectly correct/completed.\r<BR>\n" );
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document.write( "But in this problem, and in many other similar problems, there is a brilliant and cheep way to prove the uniqueness.\r<BR>\n" );
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document.write( "Notice that the function  &nbsp;&nbsp;f(x) = x*(2x)*(x+2)  &nbsp;&nbsp;is monotonic in positive domain,\r<BR>\n" );
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document.write( "since it is the product of three monotonic functions.\r<BR>\n" );
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document.write( "Therefore, &nbsp;if you guessed  &nbsp;x= 3  &nbsp;as the solution for  &nbsp;&nbsp;x*(2x)*(x+2) = 90,  &nbsp;&nbsp;you can be sure that this solution is unique\r<BR>\n" );
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document.write( "(at least, &nbsp;in the positive domain  &nbsp;x > 0 &nbsp;of real numbers).\r<BR>\n" );
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document.write( "So, the guessed solution &nbsp;x = 3 &nbsp;is a &nbsp;UNIQUE &nbsp;solution in the positive domain.\r<BR>\n" );
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document.write( "To learn this &nbsp;TRUTH &nbsp;is just a good compensation for reading this post.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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