document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=625973>Question 625973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the smallest positive integer and the largest negative integer that, by the Upper- and Lower-Bound Theorem, are upper and lower bounds for the real zeros of the polynomial function. (Enter your answers as a comma-separated list.)\r<BR>\n" );
document.write( "\n" );
document.write( "P(x) = x3 + 6x2 &#8722; 3x &#8722; 8\r<BR>\n" );
document.write( "\n" );
document.write( "x = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #394097 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=394097\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the smallest positive integer and the largest negative integer that, by the Upper- and Lower-Bound Theorem, are upper and lower bounds for the real zeros of the polynomial function. (Enter your answers as a comma-separated list.) <BR>\n" );
document.write( "P(x) = x3 + 6x2 &#8722; 3x &#8722; 8<BR>\n" );
document.write( "**<BR>\n" );
document.write( "Use rational roots theorem to solve:\r<BR>\n" );
document.write( "\n" );
document.write( "...0...|.....1......6......-3......-8<BR>\n" );
document.write( "...1...|.....1......7.......4.......-4  (Root exists between 1 & 2) (switch of signs)<BR>\n" );
document.write( "...2...|.....1......8.......13......18 (2 is upper boundary, all numbers&#8805;0<BR>\n" );
document.write( "==================<BR>\n" );
document.write( ".-1...|.....1......5.......-8........0 (-1 is a root)<BR>\n" );
document.write( "P(x)=(x+1)(x^2+5x-8)<BR>\n" );
document.write( "...0...|.....1......5.......-8<BR>\n" );
document.write( ".-2...|.....1......3.......-14<BR>\n" );
document.write( ".-3...|.....1......2.......-14<BR>\n" );
document.write( ".-4...|.....1......1.......-12<BR>\n" );
document.write( ".-5...|.....1......0.......-8<BR>\n" );
document.write( ".-6...|.....1.....-1.......-2 (Root exists between -6 & -7)(switch of signs)<BR>\n" );
document.write( ".-7...|.....1.....-2........6 (-7 is upper boundary, numbers alternate in sign)\r<BR>\n" );
document.write( "\n" );
document.write( "upper bound: x=2<BR>\n" );
document.write( "lower bound: x=-7<BR>\n" );
document.write( "real roots: x=-1, two irrational  roots between (1 & 2) and (-6 & -7) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );