document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207386>Question 1207386</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose P(x) is a quadratic polynomial (highest degree of 2) satisfying<BR>\n" );
document.write( "P(P(x)) - (P(x))^2 = x^2 + x + 2016 for all real x. Find P(x). Express the answer as a quadratic polynomial with highest degree of 2.\r<BR>\n" );
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document.write( "I've been stuck on this one for a while now and haven't found anything useful to solve. Can anyone help me out? Thanks in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845239 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=845239\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"P%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28P%28x%29%29-P%28x%29%5E2=%28a%28ax%5E2%2Bbx%2Bc%29%5E2%2Bb%28ax%5E2%2Bbx%2Bc%29%2Bc%29-%28ax%5E2%2Bbx%2Bc%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28P%28x%29%29-P%28x%29%5E2=%28a-1%29%28ax%5E2%2Bbx%2Bc%29%5E2%2Bb%28ax%5E2%2Bbx%2Bc%29%2Bc ALIGN=MIDDLE  ALT=\"P%28P%28x%29%29-P%28x%29%5E2=%28a-1%29%28ax%5E2%2Bbx%2Bc%29%5E2%2Bb%28ax%5E2%2Bbx%2Bc%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We need to have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28P%28x%29%29-P%28x%29%5E2=x%5E2%2Bx%2B2016 ALIGN=MIDDLE  ALT=\"P%28P%28x%29%29-P%28x%29%5E2=x%5E2%2Bx%2B2016\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-1%29%28ax%5E2%2Bbx%2Bc%29%5E2%2Bb%28ax%5E2%2Bbx%2Bc%29%2Bc=x%5E2%2Bx%2B2016 ALIGN=MIDDLE  ALT=\"%28a-1%29%28ax%5E2%2Bbx%2Bc%29%5E2%2Bb%28ax%5E2%2Bbx%2Bc%29%2Bc=x%5E2%2Bx%2B2016\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "On the left hand side of the equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%5E2%2Bbx%2Bc%29%5E2 ALIGN=MIDDLE  ALT=\"%28ax%5E2%2Bbx%2Bc%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is going to give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms; but there are no terms of that degree on the right hand side.  That means we must have (a-1)=0, or a=1.<br><BR>\n" );
document.write( "Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28ax%5E2%2Bbx%2Bc%29%2Bc=x%5E2%2Bx%2B2016 ALIGN=MIDDLE  ALT=\"b%28ax%5E2%2Bbx%2Bc%29%2Bc=x%5E2%2Bx%2B2016\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Equating the coefficients of x^2 on the two sides of the equation, we need to have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab=1 ALIGN=MIDDLE  ALT=\"ab=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and since a=1, b=1 also.  And now, with a=1 and b=1, we have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%2Bc%2Bc=x%5E2%2Bx%2B2016 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%2Bc%2Bc=x%5E2%2Bx%2B2016\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c=2016 ALIGN=MIDDLE  ALT=\"2c=2016\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1008 ALIGN=MIDDLE  ALT=\"c=1008\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And we have the polynomial we need.<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E2%2Bx%2B1008 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E2%2Bx%2B1008\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
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