document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=906001>Question 906001</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find a Polynomial that passes through the points (-2,-1) (-1,7) (2,-5), (3,-1)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #549586 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39620)</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=549586\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Infinitely many polynomials can contain those four points.  Pick as low a degree function as you can.\r<BR>\n" );
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document.write( "A root occurs between x at -2 and -1.<BR>\n" );
document.write( "A root occurs between x at -1 and 2.<BR>\n" );
document.write( "We do not expect a root between x at 2 and 3.\r<BR>\n" );
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document.write( "The function f may be degree 3 with a positive leading coefficient.  From -1 to 2, f decreases, and from 2 to 3, f increases.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd=y ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be the general form for the cubic equation, and specific equations can be made from each of the four given points.  Setup a system of equations and solve for a, b, c, and d.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%28-2%29%5E3%2Bb%28-2%29%5E2%2Bc%28-2%29%2Bd=-1%2Ca%28-1%29%5E3%2Bb%28-1%29%5E2%2Bc%28-1%29%2Bd=7%2Ca%5E2%5E3%2Bb%2A2%5E2%2Bc%2A2%2Bd=-5%2Ca%2A3%5E3%2Bb3%5E2%2Bc%2A3%2Bd=-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify the equations; Solve the system any way you know.\r<BR>\n" );
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document.write( "(Yes, those are FOUR different equations, but they are a SYSTEM OF EQUATION.  They are linear in the variables, a, b, c, d.  They are based on the a general form from the cubic equation which is still unknown, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd=y ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd=y\"   DISABLED_style= \"max-width:90%; height:auto;\" >). </SPAN>\n" );
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