document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=262271>Question 262271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the poynomial whose graph passes through the points (-1,1), (0,-2), (1,0), and (2,-1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #193235 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=193235\">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( "Plotting the given points looks like it might be a cubic, so let's presume a 3rd degree polynomial and see what happens.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(x)\ =\ ax^3\ +\ bx^2\ +\ cx\ +\ d\">\r<BR>\n" );
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document.write( "So if the point (-1,1) is on the graph, 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\ =\ 1\">\r<BR>\n" );
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document.write( "And if the point (0,-2) is on the graph, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(0)^3\ +\ b(0)^2\ +\ c(0)\ +\ d\ =\ -2\">\r<BR>\n" );
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document.write( "And if the point (1,0) is on the graph, 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\ =\ 0\">\r<BR>\n" );
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document.write( "And if the point (2,-1) is on the graph, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(2)^3\ +\ b(2)^2\ +\ c(2)\ +\ d\ =\ -1\">\r<BR>\n" );
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document.write( "Note that the second equation reduces to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ -2\">\r<BR>\n" );
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document.write( "Using that information and simplifying we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -a\ +\ b\ -\ c\ =\ 3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ =\ 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8a\ +\ 4b\ +\ 2c\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ -2\">\r<BR>\n" );
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document.write( "The solution to the above system is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(-\frac{4}{3},\,\frac{5}{2},\,\frac{5}{6},\,-2\right)\">\r<BR>\n" );
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document.write( "Verification of the solution of the linear system is left as an exercise for the student.\r<BR>\n" );
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document.write( "Hence the desired polynomial is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(x)\ =\ -\frac{4}{3}x^3\ +\ \frac{5}{2}x^2\ +\ \frac{5}{6}x\ -\ 2\">\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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