document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120020>Question 1120020</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use a system of equations to find the cubic function <BR>\n" );
document.write( "f(x) = ax^3 + bx^2 + cx + d that satisfies the equations. Solve the system using matrices.\r<BR>\n" );
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document.write( "f(&#8722;1) = 6<BR>\n" );
document.write( "f(1) = 4<BR>\n" );
document.write( "f(2) = 15<BR>\n" );
document.write( "f(3) = 42 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735669 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=735669\">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( "if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ ax^3\ +\ bx^2\ +\ cx\ +\ d\">\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 \ \ \ \ \ \ \ \ \ \ f(-1)\ =\ 6\">\r<BR>\n" );
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document.write( "then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(-1)^3\ +\ b(-1)^2\ +\ c(-1)\ +\ d\ =\ 6\">\r<BR>\n" );
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document.write( "Which is to say\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -a\ +\ b\ -\ c\ +\ d\ =\ 6\">\r<BR>\n" );
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document.write( "Similarly,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ =\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8a\ +\ 4b\ +\ 2c\ +\ d\ =\ 15\">\r<BR>\n" );
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document.write( "and finally,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 27a\ +\ 9b\ +\ 3c\ +\ d\ =\ 42\">\r<BR>\n" );
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document.write( "Then your augmented matrix is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \begin{bmatrix}-1\ &1\ &-1\ &1\ &\,6\ \\\,\,1\ &1\ &\,\,1\ &1\ &\,4\ \\\,\,8\ &4\ &\ 2\ &1\ &15\\27\ &9\ &\,3\ &1\ &42\\\end{bmatrix}\">\r<BR>\n" );
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document.write( "Just use Gauss-Jordan row-reduction to calculate your coefficients.  Alternatively, you can calculate the five determinants and use Cramer's Rule.  Easy-peasy if you use the MDETERM function in Excel.<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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