document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165974>Question 1165974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f(x) be a quadratic polynomial such that f(-4) = -22, f(-1)=2, and f(2)=-1. Let g(x) = f(x)^{16}. Find the sum of the coefficients of the terms in g(x) with even exponents. (For example, the sum of the coefficients of the terms in -7x^3 + 4x^2 + 10x - 5 with even exponents is (4) + (-5) = -1.)<BR>\n" );
document.write( "Please explain in detail </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #790453 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=790453\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Terrific problem.  I learned some new mathematics thinking about it and working on it.<br><BR>\n" );
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document.write( "NOTE:</b> The method I used below to solve the problem is of my own design.  It seems to me it should be valid.<br><BR>\n" );
document.write( "I will be happy to see if another tutor shows a solution that is different than mine and perhaps shows my answer to be wrong.<br><BR>\n" );
document.write( "I used an online tool to solve the problem, and its answer was VERY CLOSE to mine.  Perhaps that tool did some rounding in its calculations... or perhaps my answer was close only by chance....<br><BR>\n" );
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document.write( "We first need to find the quadratic function f(x) defined by the three given points.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "f(-4): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16a-4b%2Bc+=+-22 ALIGN=MIDDLE  ALT=\"16a-4b%2Bc+=+-22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f(-1): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-b%2Bc+=+2 ALIGN=MIDDLE  ALT=\"a-b%2Bc+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "f(2): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B2b%2Bc+=+-1 ALIGN=MIDDLE  ALT=\"4a%2B2b%2Bc+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since you are working on a problem like this, I will assume you know how to solve that system of equations, so I won't go through the details.  The function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28-3%2F2%29x%5E2%2B%281%2F2%29x%2B4 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28-3%2F2%29x%5E2%2B%281%2F2%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we need to find the function<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+%28f%28x%29%29%5E16 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28f%28x%29%29%5E16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and find the sum of the coefficients of the EVEN degree terms.<br><BR>\n" );
document.write( "This was a new problem to me.  Finding the sum of ALL the coefficients is easy -- substitute 1 for each variable and evaluate the expression.  So the sum of ALL the coefficients of g(x) would be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-3%2F2%29%2B%281%2F2%29%2B4%29%5E16+=+3%5E16+=+43046721 ALIGN=MIDDLE  ALT=\"%28%28-3%2F2%29%2B%281%2F2%29%2B4%29%5E16+=+3%5E16+=+43046721\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "But we need to find the sum of only the coefficients of the even degree terms.  How are we going to do that?<br><BR>\n" );
document.write( "Then the inspiration hit.<br><BR>\n" );
document.write( "If we change the sign of the linear term in the polynomial, we will get the same expansion except that all the terms of odd degree will have opposite signs.  So the sum<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-3%2F2%29x%5E2%2B%281%2F2%29x%2B4%29%5E16%2B%28%28-3%2F2%29x%5E2-%281%2F2%29x%2B4%29%5E16 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > [1]<br><BR>\n" );
document.write( "will have the coefficients of the even degree terms doubled and the coefficients of the odd degree terms canceled out.<br><BR>\n" );
document.write( "So the sum of the coefficients of the even degree terms will be half the sum of the coefficients of the expression [1].<br><BR>\n" );
document.write( "And we can evaluate that sum by substituting 1 for each variable everywhere.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-3%2F2%29%2B%281%2F2%29%2B4%29%5E16%2B%28%28-3%2F2%29-%281%2F2%29%2B4%29+=+3%5E16%2B2%5E16+=+43112257 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And then, finally, the sum of the coefficients of the even degree terms of g(x) is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=43112257%2F2 ALIGN=MIDDLE  ALT=\"43112257%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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document.write( "Many thanks to tutor @ikleyn for showing an easier path to the same answer I got!<br><BR>\n" );
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