document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186858>Question 1186858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following polynomial. \r<BR>\n" );
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document.write( "F(x) = x^3 + x^2 − 22x − 40\r<BR>\n" );
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document.write( "Use synthetic division to identify integer bounds of the real zeros. Find the least upper bound and the greatest lower bound guaranteed by the Upper and Lower.\r<BR>\n" );
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document.write( "Upper Bound:\r<BR>\n" );
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document.write( "Lower Bound: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849366 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the integer bounds using synthetic division:\r<BR>\n" );
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document.write( "**Understanding Upper and Lower Bound Theorem**\r<BR>\n" );
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document.write( "The Upper and Lower Bound Theorem helps us find integer values that are greater than or less than all real zeros of a polynomial.  If we perform synthetic division with a positive number *c* and all the numbers in the bottom row are positive or zero, then *c* is an *upper bound* for the real zeros. If we perform synthetic division with a negative number *c* and the numbers in the bottom row alternate in sign (positive, negative, positive, etc., or zero), then *c* is a *lower bound* for the real zeros.\r<BR>\n" );
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document.write( "**Applying Synthetic Division**\r<BR>\n" );
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document.write( "Let's test some integer values:\r<BR>\n" );
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document.write( "*   **Testing x = 6:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "6 | 1   1  -22  -40<BR>\n" );
document.write( "  |     6   42  120<BR>\n" );
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document.write( "    1   7   20   80 <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Since all the numbers in the bottom row are positive, 6 is an *upper bound*.\r<BR>\n" );
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document.write( "*   **Testing x = -5:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "-5 | 1   1  -22  -40<BR>\n" );
document.write( "   |    -5   20   10<BR>\n" );
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document.write( "     1  -4   -2  -30<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "The signs don't alternate.\r<BR>\n" );
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document.write( "*   **Testing x = -4:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "-4 | 1   1  -22  -40<BR>\n" );
document.write( "   |    -4   12   40<BR>\n" );
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document.write( "     1  -3  -10    0<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "-4 is a zero of the polynomial, which is acceptable.\r<BR>\n" );
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document.write( "*   **Testing x = -3:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "-3 | 1   1  -22  -40<BR>\n" );
document.write( "   |    -3   6   48<BR>\n" );
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document.write( "     1  -2  -16    8<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "The signs don't alternate.\r<BR>\n" );
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document.write( "*   **Testing x = -2:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "-2 | 1   1  -22  -40<BR>\n" );
document.write( "   |    -2   2   40<BR>\n" );
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document.write( "     1  -1  -20    0<BR>\n" );
document.write( "```<BR>\n" );
document.write( "-2 is also a zero of the polynomial.\r<BR>\n" );
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document.write( "*   **Testing x = -1:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "-1 | 1   1  -22  -40<BR>\n" );
document.write( "   |    -1   0   22<BR>\n" );
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document.write( "     1   0  -22  -18<BR>\n" );
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document.write( "The signs don't alternate.\r<BR>\n" );
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document.write( "**Results:**\r<BR>\n" );
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document.write( "*   **Upper Bound:** 6 (all positive or zero in bottom row)<BR>\n" );
document.write( "*   **Lower Bound:** -4 (alternating signs in bottom row, or a zero)\r<BR>\n" );
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document.write( "**Important Note:** The Upper and Lower Bound Theorem gives us *bounds* for the real zeros.  It doesn't guarantee that the bounds themselves *are* zeros. In this case, -4 and -2 are zeros, however, that does not have to be the case.<BR>\n" );
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