document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=699319>Question 699319</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a 4th degree equation, with integral coeffecients, has roots 2-i, and 1+&#8730;(3), find the equation for the polynomial. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #431320 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=431320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Both complex zeros and irrational zeros always appear in conjugate pairs.  Hence, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\ -\ i\"> is a zero, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\ +\ i\"> is also a zero.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1\ +\ \sqrt{3}\"> is a zero, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1\ -\ sqrt{3}\"> is also a zero.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha\"> is a zero of a polynomial equation, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ -\ \alpha\"> must be a factor of the polynomial.\r<BR>\n" );
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document.write( "Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ (2\ -\ i))(x\ -\ (2\ +\ i))(x\ -\ (1\ +\ \sqrt{3}))(x\ -\ (1\ -\ \sqrt{3}))\">\r<BR>\n" );
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document.write( "is the fully factored polynomial.  Multiply out the factors and set the resulting 4th degree polynomial equal to zero.  Hint:  Multiply the first two binomial factors then the last two binomial factors.  Treat the inner parentheticals as a single term.  Remember that the product of a pair of real number conjugates is the difference of two squares, and the product of a pair of complex conjugates is the sum of two squares.\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" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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