document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102896>Question 1102896</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the polynomial whose graph goes through these points? \r<BR>\n" );
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document.write( "(2,10), (-1,-4), (0,2), and (1,-2).\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717608 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717608\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "n points can be fitted with a polynomial of degree (n-1).  With 4 given points, you need a polynomial of degree 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"P%28x%29+=+ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Simply make 4 equations in the unknown coefficients a, b, c, and d using the coordinates of the given points.<BR>\n" );
document.write( "(2,10) --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B4b%2B2c%2Bd+=+10 ALIGN=MIDDLE  ALT=\"8a%2B4b%2B2c%2Bd+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [1]<BR>\n" );
document.write( "(-1,-4) --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%2Bb-c%2Bd+=+-4 ALIGN=MIDDLE  ALT=\"-a%2Bb-c%2Bd+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [2]<BR>\n" );
document.write( "(0,2) --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+2 ALIGN=MIDDLE  ALT=\"d+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [3]<BR>\n" );
document.write( "(1,-2) --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd+=+-2 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [4]<br><BR>\n" );
document.write( "Use [3] to simplify the others:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B4b%2B2c+=+8 ALIGN=MIDDLE  ALT=\"8a%2B4b%2B2c+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [5]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%2Bb-c+=+-6 ALIGN=MIDDLE  ALT=\"-a%2Bb-c+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [6]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc+=+-4 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [7]<br><BR>\n" );
document.write( "Add [6] and [7] to solve for b:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b+=+-10 ALIGN=MIDDLE  ALT=\"2b+=+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-5 ALIGN=MIDDLE  ALT=\"b+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [8]<br><BR>\n" );
document.write( "Use [8] to simplify [5] and [7]:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a-20%2B2c+=+8 ALIGN=MIDDLE  ALT=\"8a-20%2B2c+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B2c+=+28 ALIGN=MIDDLE  ALT=\"8a%2B2c+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2Bc+=+14 ALIGN=MIDDLE  ALT=\"4a%2Bc+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [9]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-5%2Bc+=+-4 ALIGN=MIDDLE  ALT=\"a-5%2Bc+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc+=+1 ALIGN=MIDDLE  ALT=\"a%2Bc+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [10]<br><BR>\n" );
document.write( "Solve [9] and [10] by elimination to solve for a and c:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+=+13 ALIGN=MIDDLE  ALT=\"3a+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+13%2F3 ALIGN=MIDDLE  ALT=\"a+=+13%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+-10%2F3 ALIGN=MIDDLE  ALT=\"c+=+-10%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The polynomial is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+%2813%2F3%29x%5E3-5x%5E2-%2810%2F3%29x%2B2 ALIGN=MIDDLE  ALT=\"p%28x%29+=+%2813%2F3%29x%5E3-5x%5E2-%2810%2F3%29x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Here is the graph...<br<<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C400%2C-3%2C3%2C-5%2C11%2C%2813%2F3%29x%5E3-5x%5E2-%2810%2F3%29x%2B2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C400%2C-3%2C3%2C-5%2C11%2C%2813%2F3%29x%5E3-5x%5E2-%2810%2F3%29x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );