document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896428>Question 896428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I determine the polynomial function of least degree whose graph passes through the given points (&#8722;1, 2), (0, 0), (1, 1), (4, 58), p(x)=?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543504 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=543504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One root of p is zero, so you can begin with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=x%28ax%5E2%2Bbx%2Bc%29 ALIGN=MIDDLE  ALT=\"p%28x%29=x%28ax%5E2%2Bbx%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to try to find a CUBIC polynomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The other three points let you form a system of three equations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=ax%5E3%2Bbx%5E2%2Bcx=p%28x%29 ALIGN=MIDDLE  ALT=\"p%28x%29=ax%5E3%2Bbx%5E2%2Bcx=p%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-<BR>\n" );
document.write( "-<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%2Bb-c=2 ALIGN=MIDDLE  ALT=\"-a%2Bb-c=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, using (-1,2).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc=1 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, using (1,1).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A4%5E3%2Bb%2A4%5E2%2Bc%2A4=58 ALIGN=MIDDLE  ALT=\"a%2A4%5E3%2Bb%2A4%5E2%2Bc%2A4=58\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64a%2B16b%2B4c=58 ALIGN=MIDDLE  ALT=\"64a%2B16b%2B4c=58\"   DISABLED_style= \"max-width:90%; height:auto;\" >, using point (4,58).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solve for the coefficients a, b, c in the system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28system%28-a%2Bb-c=2%2Ca%2Bb%2Bc=1%2C64a%2B16b%2B4c=58%29%29 ALIGN=MIDDLE  ALT=\"highlight_green%28system%28-a%2Bb-c=2%2Ca%2Bb%2Bc=1%2C64a%2B16b%2B4c=58%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C%0D%0A-1%2C1%2C-1%2C2%2C%0D%0A1%2C1%2C1%2C1%2C%0D%0A32%2C8%2C2%2C29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "R2+R1, and R3+32*R1,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C%0D%0A-1%2C1%2C-1%2C2%2C%0D%0A0%2C2%2C0%2C3%2C%0D%0A0%2C40%2C-30%2C93%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "R3-20*R2<BR>\n" );
document.write( "and one small division of R3 by constant, scaler\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C%0D%0A1%2C-1%2C1%2C-2%2C%0D%0A0%2C2%2C0%2C3%2C%0D%0A0%2C0%2C1%2C-11%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which indicates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=-11%29 ALIGN=MIDDLE  ALT=\"highlight%28c=-11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Using that continuing without further matrix operations, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=3%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28b=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=29%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28a=29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Putting this all into expected cubic p(x),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28p%28x%29=%2829%2F2%29x%5E3%2B%283%2F2%29x%5E2-11x%29 ALIGN=MIDDLE  ALT=\"highlight%28p%28x%29=%2829%2F2%29x%5E3%2B%283%2F2%29x%5E2-11x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This was not checked but it should be checked. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );