document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=188143>Question 188143</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve: x^3+3x-2x^2-6=0\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #141018 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=141018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Any possible rational zero can be found through this formula\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 -6 (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, \pm6\">\r<BR>\n" );
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document.write( "Now let's list the factors of 1 (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\">\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{2}{1}, \frac{3}{1}, \frac{6}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-3}{1}, \frac{-6}{1}\">\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 <i>possible</i> rational zeros of the function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, 2, 3, 6, -1, -2, -3, -6\">\r<BR>\n" );
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document.write( "Note: these are the possible zeros. The function may not even have rational zeros (they may be irrational or complex).\r<BR>\n" );
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document.write( "Now simply use synthetic division to find the real rational zeros\r<BR>\n" );
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document.write( "Let's see if the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >1</TD><TD >|</TD><TD >1</TD><TD >-2</TD><TD >3</TD><TD >-6</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >1</TD><TD >-1</TD><TD >2</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD >2</TD><TD >-4</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's see if the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >2</TD><TD >|</TD><TD >1</TD><TD >-2</TD><TD >3</TD><TD >-6</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >2</TD><TD >0</TD><TD >6</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >3</TD><TD >0</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2 ALIGN=MIDDLE  ALT=\"x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first three numbers in the last row 1, 0, and 3 form the coefficients to the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3 ALIGN=MIDDLE  ALT=\"x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this consequently means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6=%28x-2%29%28x%5E2%2B3%29 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6=%28x-2%29%28x%5E2%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%28x%5E2%2B3%29=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%5E2%2B3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the right side equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set each factor equal to zero\r<BR>\n" );
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document.write( "Since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is already a zero, we can ignore the first equation. So simply solve the quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the remaining solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-3 ALIGN=MIDDLE  ALT=\"x%5E2=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28-3%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-3%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides (don't forget the \"plus/minus\")\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"x=sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"x=-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i=sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solutions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2%2B3x-6 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2%2B3x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"x=sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"x=-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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