document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=331300>Question 331300</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the polynomial x^3+8x^2+19x+12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #237508 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=237508\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Factor the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B8x%5E2%2B19x%2B12 ALIGN=MIDDLE  ALT=\"x%5E3%2B8x%5E2%2B19x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the highest degree is 3, there exist 3 roots to this poynomial\r<BR>\n" );
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document.write( "The real roots (there might be some complex roots) will be from the set <BR>\n" );
document.write( " -/+1, -/+2, -/+3, -/+4,-/+6, -/+12\r<BR>\n" );
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document.write( "Use Discarte's sign rule<BR>\n" );
document.write( "Number of positive roots P(x) ++++ no sign change, so no positive roots<BR>\n" );
document.write( "number of negative roots P(-x) -+-+ 3 sign changes, so either 3 negative roots or 1 negative root and 2 complex roots\r<BR>\n" );
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document.write( "Since we know that at least one of the roots is negative, the roots are from <BR>\n" );
document.write( "the set  -1,-2,-3,-4,-6,-12<BR>\n" );
document.write( "so experiment with -1 and see  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-1%29=%28-1%29%5E3+%2B+8%28-1%29%5E2%2B19%28-1%29+%2B12+=+-1%2B8-19%2B12=0 ALIGN=MIDDLE  ALT=\"P%28-1%29=%28-1%29%5E3+%2B+8%28-1%29%5E2%2B19%28-1%29+%2B12+=+-1%2B8-19%2B12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so -1 is a root.\r<BR>\n" );
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document.write( "Now apply either long division or synthetic division to reduce the polynomial by one degree yield  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x%2B12 ALIGN=MIDDLE  ALT=\"x%5E2%2B7x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > this can be easily factored into (x+3)(x+4)\r<BR>\n" );
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document.write( "so the original polymial can be factord into <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B8x%5E2%2B19x%2B12+=+%28x%2B1%29%2A%28x%2B3%29%2A%28x%2B4%29 ALIGN=MIDDLE  ALT=\"x%5E3%2B8x%5E2%2B19x%2B12+=+%28x%2B1%29%2A%28x%2B3%29%2A%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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