document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=295818>Question 295818</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The complex number 2+i is a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-z%5E2-pz%2B15=0 ALIGN=MIDDLE  ALT=\"z%5E3-z%5E2-pz%2B15=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where p is real. Explain why 2-i is another root and find the third root. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #213264 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=213264\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Because complex roots ALWAYS come in conjugate pairs.  Therefore if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\ +\ \beta i\"> is a root of a polynomial equation, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\ -\ \beta i\"> is also a root of that same polynomial equation.  Guaranteed.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \gamma\"> is the root of a polynomial equation, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x\ -\ \gamma)\"> is a factor of the polynomial.\r<BR>\n" );
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document.write( "Since you know two of the roots of the equation, you know two of the factors of the polynomial, namely:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (z\ -\ (2\ +\ i))\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (z\ -\ (2\ -\ i))\">\r<BR>\n" );
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document.write( "So, step one is to multiply these two factors together.  I'll leave that as an exercise for the student.  Hint:  The product of two conjugates is the difference of two squares.  And don't forget <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i^2\ =\ -1\">\r<BR>\n" );
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document.write( "The next step is to divide the original polynomial by the quadratic trinomial you just derived in the previous step.  Use polynomial long division.  See http://www.purplemath.com/modules/polydiv2.htm if you need a refresher on the process.  You will need to select an appropriate value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> so that you end up with a zero remainder.  The quotient you obtain will be the third factor.\r<BR>\n" );
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document.write( "Good luck.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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