document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000064>Question 1000064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve the question <BR>\n" );
document.write( "if ax^3+bx^2+cx+d is divisible by ax^2+c then a,b,c,d are in ap or gp or hp or none? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #617579 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=617579\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In general, the answer is NONE.<BR>\n" );
document.write( "Adding some requirements to the coefficients a, b, c, and d,<BR>\n" );
document.write( "we could be sure to have a gp,<BR>\n" );
document.write( "or even a gp that is also an ap, and a hp.\r<BR>\n" );
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document.write( "If there are no specifications about a,b,c,and d,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=b%2Cc=d=0%29 ALIGN=MIDDLE  ALT=\"system%28a=b%2Cc=d=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then a,b,c,d would be neither an ap, nor a gp, nor a hp.\r<BR>\n" );
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document.write( "If it were specified that none of the coefficients a, b,c,and d is zero,<BR>\n" );
document.write( "when we divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > we find<BR>\n" );
document.write( "a quotient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bb%2Fa ALIGN=MIDDLE  ALT=\"x%2Bb%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "a remainder of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d-bc%2Fa ALIGN=MIDDLE  ALT=\"d-bc%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In that case,<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d-bc%2Fa=0 ALIGN=MIDDLE  ALT=\"d-bc%2Fa=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=bc%2Fa ALIGN=MIDDLE  ALT=\"d=bc%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fb=c%2Fa ALIGN=MIDDLE  ALT=\"d%2Fb=c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Conversely, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fb=c%2Fa ALIGN=MIDDLE  ALT=\"d%2Fb=c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=bc%2Fa ALIGN=MIDDLE  ALT=\"d=bc%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ad=bc ALIGN=MIDDLE  ALT=\"ad=bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fc=b%2Fa ALIGN=MIDDLE  ALT=\"d%2Fc=b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then the remainder is zero, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fb=c%2Fa ALIGN=MIDDLE  ALT=\"d%2Fb=c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=bc%2Fa ALIGN=MIDDLE  ALT=\"d=bc%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > to happen,<BR>\n" );
document.write( "it is not necessary that a,b,c,d form an ap, an gp, or an hp.<BR>\n" );
document.write( "Sometimes a,b,c,d will be an ap, a gp, and a hp,<BR>\n" );
document.write( "sometimes it will only be a gp,<BR>\n" );
document.write( "but other times it will be none of those.<BR>\n" );
document.write( "For example, 1,2,3,6 is neither an ap, nor a gp, nor a hp.\r<BR>\n" );
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document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fb=c%2Fa ALIGN=MIDDLE  ALT=\"d%2Fb=c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=bc%2Fa ALIGN=MIDDLE  ALT=\"d=bc%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%28ac%29 ALIGN=MIDDLE  ALT=\"b=sqrt%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa=sqrt%28c%2Fa%29 ALIGN=MIDDLE  ALT=\"b%2Fa=sqrt%28c%2Fa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "a,b,c,d is a gp with common ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28c%2Fa%29 ALIGN=MIDDLE  ALT=\"sqrt%28c%2Fa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "1,2,4,8 is an example.<BR>\n" );
document.write( "A special case of that is when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b=c=d ALIGN=MIDDLE  ALT=\"a=b=c=d\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which makes a,b,c,d<BR>\n" );
document.write( "a gp, an ap, and a hp. </SPAN>\n" );
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