document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191261>Question 191261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me factorize and state the roots of the following polynomial. Please, I beg you, I need to know how to to this and would really appreciate whatever help I can get. This was a take home assignment. Thanks.\r<BR>\n" );
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document.write( "P(x)=4x^7+28x^6+27x^5-203x^4-591x^3-431x^2+146x+120 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143585 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=143585\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Wow, this is as long as it gets (well I hope so anyway...). Here are two ways to do this:\r<BR>\n" );
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document.write( "Method #1: \r<BR>\n" );
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document.write( "Use the rational root theorem to find all of the <i>possible</i> roots and test EVERY possible root to see if it is actually a root. Since the degree of the equation is 7, this means that once you find 7 roots, then you don't need to check any more possible roots.\r<BR>\n" );
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document.write( "Here's where this method gets really tedious: there are A LOT of possible roots (48 in total)\r<BR>\n" );
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document.write( "But, here's how to find them:\r<BR>\n" );
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document.write( "Any rational zero can be found through this equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the factors of the last and first coefficients\r<BR>\n" );
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document.write( "So let's list the factors of 120 (the last coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm2, \pm3, \pm4, \pm5, \pm6, \pm8, \pm10, \pm12, \pm15, \pm20, \pm24, \pm30, \pm40, \pm60, \pm120\">\r<BR>\n" );
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document.write( "Now let's list the factors of 4 (the first coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm2, \pm4\">\r<BR>\n" );
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document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{2}{1}, \frac{2}{2}, \frac{2}{4}, \frac{3}{1}, \frac{3}{2}, \frac{3}{4}, \frac{4}{1}, \frac{4}{2}, \frac{4}{4}, \frac{5}{1}, \frac{5}{2}, \frac{5}{4}, \frac{6}{1}, \frac{6}{2}, \frac{6}{4}, \frac{8}{1}, \frac{8}{2}, \frac{8}{4}, \frac{10}{1}, \frac{10}{2}, \frac{10}{4}, \frac{12}{1}, \frac{12}{2}, \frac{12}{4}, \frac{15}{1}, \frac{15}{2}, \frac{15}{4}, \frac{20}{1}, \frac{20}{2}, \frac{20}{4}, \frac{24}{1}, \frac{24}{2}, \frac{24}{4}, \frac{30}{1}, \frac{30}{2}, \frac{30}{4}, \frac{40}{1}, \frac{40}{2}, \frac{40}{4}, \frac{60}{1}, \frac{60}{2}, \frac{60}{4}, \frac{120}{1}, \frac{120}{2}, \frac{120}{4}, -\frac{1}{1}, -\frac{1}{2}, -\frac{1}{4}, -\frac{2}{1}, -\frac{2}{2}, -\frac{2}{4}, -\frac{3}{1}, -\frac{3}{2}, -\frac{3}{4}, -\frac{4}{1}, -\frac{4}{2}, -\frac{4}{4}, -\frac{5}{1}, -\frac{5}{2}, -\frac{5}{4}, -\frac{6}{1}, -\frac{6}{2}, -\frac{6}{4}, -\frac{8}{1}, -\frac{8}{2}, -\frac{8}{4}, -\frac{10}{1}, -\frac{10}{2}, -\frac{10}{4}, -\frac{12}{1}, -\frac{12}{2}, -\frac{12}{4}, -\frac{15}{1}, -\frac{15}{2}, -\frac{15}{4}, -\frac{20}{1}, -\frac{20}{2}, -\frac{20}{4}, -\frac{24}{1}, -\frac{24}{2}, -\frac{24}{4}, -\frac{30}{1}, -\frac{30}{2}, -\frac{30}{4}, -\frac{40}{1}, -\frac{40}{2}, -\frac{40}{4}, -\frac{60}{1}, -\frac{60}{2}, -\frac{60}{4}, -\frac{120}{1}, -\frac{120}{2}, -\frac{120}{4}\">\r<BR>\n" );
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document.write( "Now simplify\r<BR>\n" );
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document.write( "These are all the distinct rational zeros of the function that could occur\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, \frac{1}{2}, \frac{1}{4}, 2, 3, \frac{3}{2}, \frac{3}{4}, 4, 5, \frac{5}{2}, \frac{5}{4}, 6, 8, 10, 12, 15, \frac{15}{2}, \frac{15}{4}, 20, 24, 30, 40, 60, 120, -1, -\frac{1}{2}, -\frac{1}{4}, -2, -3, -\frac{3}{2}, -\frac{3}{4}, -4, -5, -\frac{5}{2}, -\frac{5}{4}, -6, -8, -10, -12, -15, -\frac{15}{2}, -\frac{15}{4}, -20, -24, -30, -40, -60, -120\">\r<BR>\n" );
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document.write( "Once you have all of the <i>possible</i> rational roots, either plug them in directly or use synthetic division to determine which ones are actually roots. \r<BR>\n" );
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document.write( "<hr>\r<BR>\n" );
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document.write( "Method #2:\r<BR>\n" );
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document.write( "Since the first method seems like a lot of busy work (which it is), just use a graphing calculator to find some of the roots, and then use those roots to find other roots (using synthetic division). Eventually, you'll reduce that massive polynomial to a quadratic which you can solve using the quadratic formula.\r<BR>\n" );
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document.write( "If you're completely stuck, then repost or email me. By the way, the 7 roots are: 3, -1/2, 1/2, -4, -2, -2+i, -2-i\r<BR>\n" );
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document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E7%2B28x%5E6%2B27x%5E5-203x%5E4-591x%5E3-431x%5E2%2B146x%2B120 ALIGN=MIDDLE  ALT=\"4x%5E7%2B28x%5E6%2B27x%5E5-203x%5E4-591x%5E3-431x%5E2%2B146x%2B120\"   DISABLED_style= \"max-width:90%; height:auto;\" > completely factors (over the reals) to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%282x%2B1%29%282x-1%29%28x%2B4%29%28x%2B2%29%28x%5E2%2B4x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-3%29%282x%2B1%29%282x-1%29%28x%2B4%29%28x%2B2%29%28x%5E2%2B4x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E7%2B28x%5E6%2B27x%5E5-203x%5E4-591x%5E3-431x%5E2%2B146x%2B120=%28x-3%29%282x%2B1%29%282x-1%29%28x%2B4%29%28x%2B2%29%28x%5E2%2B4x%2B5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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