document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1131984>Question 1131984</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Find an equation for f(x), the polynomial of smallest degree with real coefficients such that f(x) breaks through the x-axis at &#8722;4, breaks through the x-axis at 5, has complex roots of &#8722;3&#8722;i and 5+5i and passes through the point (0,&#8722;54).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #748765 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=748765\">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( "Corrolaries to the Fundamental Theorem of Algebra tell us the following things:\r<BR>\n" );
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document.write( "1. A polynomial function with complex coefficients of degree <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> has exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> complex zeros.\r<BR>\n" );
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document.write( "2. There is always an even number of complex zeros with non-zero imaginary parts, and these zeros always appear as conjugate pairs.  That is if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\,+\,bi\"> is a zero of a polynomial function with complex coefficients, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\,-\,bi\"> is also a zero of that polynomial.\r<BR>\n" );
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document.write( "Both of these assertions apply to polynomials with strictly real coefficients because real numbers are the subset of the complex numbers where the imaginary part is zero.\r<BR>\n" );
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document.write( "We also know that if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large z\ =\ a\,\pm\,bi\"> is a zero of a polynomial function, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ -\ z\"> is a factor of the polynomial.  Further, any such polynomial has an additional non-zero constant factor <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\">.  For example, the general third-degree polynomial function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho_3(x)\ =\ a_1x^3\ +\ a_2x^2\ +\ a_3x\ +\ a_4\"> has the following factors: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha(x\ -\ z_1)(x\ -\ z_2)(x\ -\ z_3)\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha = a_1\">\r<BR>\n" );
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document.write( "With the given information, we can identify six roots of the desired polynomial, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -4,\ 5,\ -3\,-\,i,\ -3\,+\,i,\ 5\,+\,5i,\ 5\,-\,5i\">\r<BR>\n" );
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document.write( "Hence, the polynomial of the smallest degree that has these zeros is a polynomial of degree six.\r<BR>\n" );
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document.write( "Next, we can identify the seven factors of our desired polynomial:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \alpha,\ (x\,+\,4),\ (x\,-\,5),\ (x\,+\,3\,-\,i),\ (x\,+\,3\,+\,i)\ (x\,-\,5\,-\,5i),\ (x\,-\,5\,+\,5i)\">\r<BR>\n" );
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document.write( "Multiplying the factors in pairs:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ +\ 4)(x\ -\ 5)\ =\ x^2\ -\ x\ +\ 20\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\,+\,3\,-\,i)(x\,+\,3\,+\,i)\ =\ x^2 +\ 3x\ +\ xi\ +\ 3x\ +\ 9\ +\ 3i\ -\ xi\ -\ 3i\ -\ (-1)\ =\ x^2\ +\ 6x\ +\ 10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\,-\,5\,-\,5i)(x\,-\,5\,+\,5i)\ =\ x^2\ -\ 5x\ +\ 5ix\ -\ 5x\ +\ 25\ -\ 25i\ -\ 5ix\ +\ 25i\ -\ (-1)25\ =\ x^2\ -\ 10x\ +\ 50)\">\r<BR>\n" );
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document.write( "So far we have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho(x)\ =\ \alpha(x^2\ -\ x\ -\ 20)(x^2\ +\ 6x\ +\ 10)(x^2\ -\ 10x\ +\ 50)\">\r<BR>\n" );
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document.write( "But we are given that the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,-54)\"> is on the graph.  Since any point on the graph is of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(c,\rho(c)\)\"> we can see that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho(0)\ =\ -54\">, hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(0)\ =\ \alpha\((0)^2\ -\ (0)\ -\ 20\)\((0)^2\ +\ 6(0)\ +\ 10\)\((0)^2\ -\ 10(0)\ +\ 50\)\ =\ \alpha(-20)(10)(50)\ =\ -54\">\r<BR>\n" );
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document.write( "Solving for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  -10000\alpha\ =\ -54\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \alpha\ =\ 0.0054\">\r<BR>\n" );
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document.write( "And now we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ 0.0054(x^2\ -\ x\ -\ 20)(x^2\ +\ 6x\ +\ 10)(x^2\ -\ 10x\ +\ 50)\">\r<BR>\n" );
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document.write( "The last step is multiplying the three quadratic trinomials and distributing the lead coefficient to derive the final form of the sixth-degree polynomial function in standard form.  I leave this as an exercise for the student.<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( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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