document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823760>Question 823760</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate A = (a^2 &#8722; 3a +1)^2 &#8722; (1&#8722; a)(&#8722;2a +1)(1&#8722; 3a) . Then WITHOUT doing any<BR>\n" );
document.write( "further calculation write down the value of B = (a^2 + 3a +1)^2 &#8722; (1+ a)(2a +1)(1+ 3a) . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495899 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495899\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Without doing any calculation we can see that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28a%29 ALIGN=MIDDLE  ALT=\"A%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28a%29 ALIGN=MIDDLE  ALT=\"B%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both polynomial functions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > with degree 4.\r<BR>\n" );
document.write( "\n" );
document.write( "Without doing much calculation we can see that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-a%29=%28%28-a%29%5E2-3%28-a%29%2B1%29%5E2-%281-%28-a%29%29%28-2%28-a%29%2B1%29%281-3%28-a%29%29=%28a%5E2%2B3a%2B1%29%5E2-%281%2Ba%29%282a%2B1%29%281%2B3a%29=B%28a%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28-a%29=A%28a%29 ALIGN=MIDDLE  ALT=\"B%28-a%29=A%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "So, the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-a%29=B%28a%29 ALIGN=MIDDLE  ALT=\"A%28-a%29=B%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes calculations easier.<BR>\n" );
document.write( "We do not even need to calculate to see that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%280%29=A%28-0%29=B%280%29 ALIGN=MIDDLE  ALT=\"A%280%29=A%28-0%29=B%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can calculate the value of the functions for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-0.5 ALIGN=MIDDLE  ALT=\"a=-0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.5 ALIGN=MIDDLE  ALT=\"a=0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%280%29=%280%5E2%2B3%2A0%2B1%29%5E2-%281%2B0%29%281%2B2%2A0%29%281%2B3%2A0%29=1%5E2-1%2A1%2A1=1-1=0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28-1%29=A%281%29=%281%5E2-3%2A1%2B1%29%5E2-%281-1%29%281-2%29%281-3%29=%281-3%2B1%29%5E2-0%2A%28-1%29%2A%28-2%29=%28-1%29%5E2-1=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-1%29=B%281%29=%281%2B3%2B1%29%5E2-%281%2B1%29%281%2B2%29%281%2B3%29=5%5E2-2%2A3%2A4=25-24=1%0D%0A%7B%7B%7BA%280.5%29=%280.5%5E2-3%2A0.5%2B1%29%5E2-%281-0.5%29%281-1%29%281-1.5%29=%280.25-1.5%2B1%29%5E2-0.5%2A0%2A%28-0.5%29=%28-0.25%29%5E2-0=0.0625 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-0.5%29=B%280.5%29=%280.5%5E2%2B3%2A0.5%2B1%29%5E2%5E2-%281%2B0.5%29%281%2B1%29%281%2B1.5%29=%280.25%2B1.5%2B1%29%5E2-1.5%2A2%2A2.5=2.75%5E2-7.5=7.5625-7.5=0.0625 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In sum<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%280%29=B%280%29=0 ALIGN=MIDDLE  ALT=\"A%280%29=B%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%281%29=A%28-1%29=B%281%29=B%28-1%29=1 ALIGN=MIDDLE  ALT=\"A%281%29=A%28-1%29=B%281%29=B%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%280.5%29=A%28-0.5%29=B%280.5%29=B%28-0.5%29=0.0625 ALIGN=MIDDLE  ALT=\"A%280.5%29=A%28-0.5%29=B%280.5%29=B%28-0.5%29=0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We know that the polynomial of degree 4 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28a%29=a%5E4 ALIGN=MIDDLE  ALT=\"P%28a%29=a%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > has<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=0 ALIGN=MIDDLE  ALT=\"P%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%29=P%28-1%29=1 ALIGN=MIDDLE  ALT=\"P%281%29=P%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280.5%29=P%28-0.5%29=0.0625 ALIGN=MIDDLE  ALT=\"P%280.5%29=P%28-0.5%29=0.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and since 5 points uniquely determine a degree 4 polynomial,<BR>\n" );
document.write( "just like 2 points uniquely determine a line,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28a%29=B%28a%29=a%5E4 ALIGN=MIDDLE  ALT=\"A%28a%29=B%28a%29=a%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );