document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=467123>Question 467123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the degree of a polynomial p(x)is no larger than 3. we know that p(0)=3, <BR>\n" );
document.write( "P(-x)=p(x), and p(1/x)=p(x)/x2. find p(1)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #320411 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=320411\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let M = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Then M + 461 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Subtract the top equation from the bottom equation, to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=461+=+%28x%2B1%29%5E2+-+x%5E2 ALIGN=MIDDLE  ALT=\"461+=+%28x%2B1%29%5E2+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=461+=+x%5E2+%2B+2x+%2B+1+-+x%5E2 ALIGN=MIDDLE  ALT=\"461+=+x%5E2+%2B+2x+%2B+1+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> 461 = 2x + 1<BR>\n" );
document.write( "==> 2x = 460\r<BR>\n" );
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document.write( "==> x = 230.\r<BR>\n" );
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document.write( "Hence M = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=230%5E2 ALIGN=MIDDLE  ALT=\"230%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 52,900.\r<BR>\n" );
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document.write( "We must have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=deg+p%28x%29+%3C=+3 ALIGN=MIDDLE  ALT=\"deg+p%28x%29+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "p(0) = 3, p(-x) = p(x), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%281%2Fx%29+=+p%28x%29%2Fx%5E2 ALIGN=MIDDLE  ALT=\"p%281%2Fx%29+=+p%28x%29%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+ax%5E3+%2Bbx%5E2+%2B+cx+%2B+d ALIGN=MIDDLE  ALT=\"p%28x%29+=+ax%5E3+%2Bbx%5E2+%2B+cx+%2B+d\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> p(0) = d = 3, as given.<BR>\n" );
document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%281%2Fx%29+=+a%2Fx%5E3+%2B+b%2Fx%5E2+%2B+c%2Fx+%2B+d+=+ax+%2B+b+%2B+c%2Fx+%2B+d%2Fx%5E2+=+p%28x%29%2Fx%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> a = 0 by direct correspondence of terms, <BR>\n" );
document.write( "b = d, and c is some real number.\r<BR>\n" );
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document.write( "==> b = d = 3, and a = 0.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+3x%5E2+%2B+cx+%2B+3 ALIGN=MIDDLE  ALT=\"p%28x%29+=+3x%5E2+%2B+cx+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now p(x) = p(-x) ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B+cx+%2B+3+=+3x%5E2+-cx+%2B+3 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+cx+%2B+3+=+3x%5E2+-cx+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> cx = -cx ==> 2cx = 0.\r<BR>\n" );
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document.write( "Since x is not zero for all values of x in the domain of p(x), this means that c = 0.\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+3x%5E2+%2B+3 ALIGN=MIDDLE  ALT=\"p%28x%29+=+3x%5E2+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%281%29+=+3%2A1%5E2+%2B+3+=+3+%2B+3+=+6 ALIGN=MIDDLE  ALT=\"p%281%29+=+3%2A1%5E2+%2B+3+=+3+%2B+3+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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