document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316701>Question 316701</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find an nth-degree polynomial function with real coefficients satisfying the given conditions? n=3; 1 and 5i are zeros; f (-1)=-104 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226638 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=226638\">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( "Use the following facts:\r<BR>\n" );
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document.write( "The Fundamental Theorem of Algebra, namely that every <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\">-th degree polynomial function has exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> zeros, counting all multiplicities.\r<BR>\n" );
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document.write( "Complex roots ALWAYS come in conjugate pairs.  That means that if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0\ +\ 5i\"> is a zero, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0\ -\ 5i\"> is also a zero of the desired polynomial function.\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 function in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ -\ \alpha\"> is a factor of the polynomial.\r<BR>\n" );
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document.write( "A family of polynomial functions of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ p(x)\ =\ k\left(a_0x^n\ +\ a_1x^{n-1}\ +\ \cdots\ +\ a_ix^{n-i}\ +\ \cdots\ +\ a_{n-1}x\ +\ a_n\right)\">\r<BR>\n" );
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document.write( "all have the same zeros.\r<BR>\n" );
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document.write( "So we know the following things:\r<BR>\n" );
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document.write( "1.  The desired polynomial function has exactly 3 zeros.\r<BR>\n" );
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document.write( "2.  The zeros are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5i\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -5i\">.\r<BR>\n" );
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document.write( "3.  The factors of the polynomial, including any possible common factor multiplier are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k(x\ -\ 1)(x\ -\ 5i)(x\ +\ 5i)\">\r<BR>\n" );
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document.write( "Multiply the factors:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ p(x)\ =\ k(x\ -\ 1)(x^2\ +\ 25)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ p(x)\ =\ k(x^3\ -\ x^2\ +\ 25x\ -\ 25)\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(-1)\ =\ -104\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k((-1)^3\ -\ (-1)^2\ +\ 25(-1)\ -\ 25)\ =\ k\,\cdot\,(-52)\">\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ 2\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ 2x^3\ -\ 2x^2\ +\ 50x\ -\ 50\">\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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