document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=499169>Question 499169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the greatest lower bound and the smallest upper bound of f(x)=x^4-x^3-12x^2-2x+3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #337631 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=337631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the greatest lower bound and the smallest upper bound of <BR>\n" );
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document.write( "f(x)=x^4-x^3-12x^2-2x+3<BR>\n" );
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document.write( "....B...<BR>\n" );
document.write( "....0..|......1....-1......-12........-2..........3......<BR>\n" );
document.write( "....1..|......1......0......-12........-14.....-11.....<BR>\n" );
document.write( "....2..|..... 1......1......-10........-22.....-41....<BR>\n" );
document.write( "....3..|......1......2.......-6.........-20.....-57....<BR>\n" );
document.write( "....4..|......1......3.........0.........-2.......-5.....<BR>\n" );
document.write( "....5..|......1......4.........8..........38.......187...(x=5 is smallest upper bound)<BR>\n" );
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document.write( "....0..|......1....-1......-12........-2..........3......<BR>\n" );
document.write( "..-1..|......1....-2......-10..........8.........-5.....<BR>\n" );
document.write( "..-2..|......1....-3.......-6..........10.......-17....<BR>\n" );
document.write( "..-3..|......1....-4.........0.........-2......... 9......(x=-3 is the greatest lower bound)<BR>\n" );
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document.write( "Upper and Lower Bound Theorem for Real Number Roots.<BR>\n" );
document.write( "Using synthetic division to divide f(x) by x-B<BR>\n" );
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document.write( "If B>0 and all numbers in the last row from the synthetic division by x-B are positive or zero,<BR>\n" );
document.write( "then, B is the smallest upper bound.<BR>\n" );
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document.write( "If B<0 and all numbers in the last row from the synthetic division by x-B alternate in sign<BR>\n" );
document.write( "then, B is the greatest lower bound. Zero can be considered positive or negative. </SPAN>\n" );
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