document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189444>Question 1189444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of all integral values of x for which (18(x^2 - 7x + 10)/x^3 - 6x^2 + 3x + 10) has an integral value is\r<BR>\n" );
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document.write( "A) -17<BR>\n" );
document.write( "B) -19<BR>\n" );
document.write( "C) -12<BR>\n" );
document.write( "D) -14<BR>\n" );
document.write( "E) -20 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820803 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=820803\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Your parentheses are in the wrong places.<br><BR>\n" );
document.write( "But at least, unlike a large number of people who submit questions to this forum, you TRIED to use parentheses....<br><BR>\n" );
document.write( "Here is the expression you posted: (18(x^2 - 7x + 10)/x^3 - 6x^2 + 3x + 10) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818%28x%5E2+-+7x+%2B+10%29%2Fx%5E3+-+6x%5E2+%2B+3x+%2B+10%29 ALIGN=MIDDLE  ALT=\"%2818%28x%5E2+-+7x+%2B+10%29%2Fx%5E3+-+6x%5E2+%2B+3x+%2B+10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Undoubtedly the expression you intended to show is this: 18(x^2 - 7x + 10)/(x^3 - 6x^2 + 3x + 10) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%28x%5E2+-+7x+%2B+10%29%2F%28x%5E3+-+6x%5E2+%2B+3x+%2B+10%29 ALIGN=MIDDLE  ALT=\"18%28x%5E2+-+7x+%2B+10%29%2F%28x%5E3+-+6x%5E2+%2B+3x+%2B+10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Factor the numerator and denominator and try to simplify the expression.<br><BR>\n" );
document.write( "The numerator factors easily....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%28%28x-5%29%28x-2%29%29%2F%28x%5E3+-+6x%5E2+%2B+3x+%2B+10%29 ALIGN=MIDDLE  ALT=\"18%28%28x-5%29%28x-2%29%29%2F%28x%5E3+-+6x%5E2+%2B+3x+%2B+10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are many formal ways to factor the cubic polynomial in the denominator.  However, since we are hoping the expression will simplify, let's check specifically to see if (x-5) and/or (x-2) are factors of the denominator.<br><BR>\n" );
document.write( "In fact they are both factors; the expression simplifies to<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%28%28x-5%29%28x-2%29%29%2F%28%28x-5%29%28x-2%29%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"18%28%28x-5%29%28x-2%29%29%2F%28%28x-5%29%28x-2%29%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The function is undefined for values that make the denominator 0: 5, 2, and -1.<br><BR>\n" );
document.write( "In this problem, we are not concerned with where the function is undefined; we are interested in the values of x that make the expression have an integer value.  So we can cancel the common factors in the numerator and denominator to simplify the expression to<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"18%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That expression will have an integer value whenever (x+1) is a (positive or negative) factor of 18.  We can simply make a list:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   x+1  x\r\n" );
document.write( "  --------\r\n" );
document.write( "    18  17\r\n" );
document.write( "     9   8\r\n" );
document.write( "     6   5\r\n" );
document.write( "     3   2\r\n" );
document.write( "     2   1\r\n" );
document.write( "     1   0\r\n" );
document.write( "    -1  -2\r\n" );
document.write( "    -2  -3\r\n" );
document.write( "    -3  -4\r\n" );
document.write( "    -6  -7\r\n" );
document.write( "    -9 -10\r\n" );
document.write( "   -18 -19</PRE><BR>\n" );
document.write( "Adding the values of x in the second column gives us a total of -12.<br><BR>\n" );
document.write( "ANSWER: C) -12<br><BR>\n" );
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