document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=973934>Question 973934</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)  if x is a rational number and (x+1)^3 -(x-1)^3 / (x+1)^2 - (x-1)^2 = 2, then the sum of numerator and denominator of x is ?\r<BR>\n" );
document.write( "\n" );
document.write( "2)  if x^2 + y^2 + 1 = 2x, then x^3 + y^5 = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #596147 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=596147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 2)  If x^2+y^2+1=2x, then<BR>\n" );
document.write( "x^2+y^2+1-2x=0 ---> (x^2-2x+1)+y^2=0 ---> (x-1)^2+y^2=0 ---> y=0 and x-1=0<-->x=1.<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "x^3 + y^5 = 1^2 + 0^2 =1 + 0 = 1<BR>\n" );
document.write( " <BR>\n" );
document.write( "1) Did you really mean<BR>\n" );
document.write( "(x+1)^3 -(x-1)^3 / (x+1)^2 - (x-1)^2 =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= 2?<BR>\n" );
document.write( "Or did you mean<BR>\n" );
document.write( "[ (x+1)^3 -(x-1)^3 ] / [ (x+1)^2 - (x-1)^2 ]=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= 2?<BR>\n" );
document.write( "I like the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > expression,<BR>\n" );
document.write( "because it has a sort of elegant symmetry, and simplifies nicely.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can expand all those cubes and squares and laboriously simplify.<BR>\n" );
document.write( "We can also use special products to simplify the expression with less risk for mistakes.<BR>\n" );
document.write( "If we make the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=x%2B1%2Cb=x-1%29 ALIGN=MIDDLE  ALT=\"system%28a=x%2B1%2Cb=x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to save some writing, we can simplify easier.<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3-b%5E3=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2=%28a-b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2=%28a-b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E3-b%5E3%29%2F%28a%5E2-b%5E2%29=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29%2F%28a-b%29%28a%2Bb%29=%28a%5E2%2Bab%2Bb%5E2%29%2F%28a%2Bb%29=%28%28a%5E2%2B2ab%2Bb%5E2%29-ab%29%2F%28a%2Bb%29=%28%28a%2Bb%29%5E2-ab%29%2F%28a%2Bb%29=%28a%2Bb%29%5E2%2F%28a%2Bb%29-ab%29%2F%28a%2Bb%29=a%2Bb-ab%2F%28a%2Bb%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, going back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%2B%28x-1%29-%28x%2B1%29%28x-1%29%2F%28%28x%2B1%29%2B%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%2B%28x-1%29-%28x%2B1%29%28x-1%29%2F%28%28x%2B1%29%2B%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-%28x%5E2-1%29%2F%282x%29=%284x%5E2-%28x%5E2-1%29%29%2F%222+x%22 ALIGN=MIDDLE  ALT=\"2x-%28x%5E2-1%29%2F%282x%29=%284x%5E2-%28x%5E2-1%29%29%2F%222+x%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2-x%5E2%2B1%29%2F%222+x%22=%283x%5E2%2B1%29%2F%222+x%22 ALIGN=MIDDLE  ALT=\"%284x%5E2-x%5E2%2B1%29%2F%222+x%22=%283x%5E2%2B1%29%2F%222+x%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29=%283x%5E2%2B1%29%2F%222+x%22 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , so if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28%28x%2B1%29%5E3+-%28x-1%29%5E3%29%2F%28%28x%2B1%29%5E2+-+%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E2%2B1%29%2F%222+x%22=2 ALIGN=MIDDLE  ALT=\"%283x%5E2%2B1%29%2F%222+x%22=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B1=4x ALIGN=MIDDLE  ALT=\"3x%5E2%2B1=4x\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-4x%2B1=0 ALIGN=MIDDLE  ALT=\"3x%5E2-4x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-1%29%28x-1%29=0 ALIGN=MIDDLE  ALT=\"%283x-1%29%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=1%2F3%2C%22or%22%2Cx=1=1%2F1%29 ALIGN=MIDDLE  ALT=\"system%28x=1%2F3%2C%22or%22%2Cx=1=1%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, the sum of numerator and denominator of x is either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B3=4 ALIGN=MIDDLE  ALT=\"1%2B3=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2=%28%28x%2B1%29%5E3%28x%2B1%29%5E2+-%28x-1%29%5E3++-+%28x-1%29%5E2+%28x%2B1%29%5E2%29%2F%28x%2B1%29%5E2 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=%28%28x%2B1%29%5E5-%28x-1%29%5E3-%28x%5E2-1%29%5E2%29%2F%28x%2B1%29%5E2=%28%28x%5E5%2B5x%5E4%2B10x%5E3%2B10x%5E2%2B5x%2B1%29-%28x%5E3-3x%5E2%2B3x-1%29-%28x%5E4-2x%5E2%2B1%29%29%2F%28x%2B1%29%5E2 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=%28x%5E5%2B5x%5E4%2B10x%5E3%2B10x%5E2%2B5x%2B1-x%5E3%2B3x%5E2-3x%2B1-x%5E4%2B2x%5E2-1%29%2F%28x%2B1%29%5E2+=%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29%2F%28x%5E2%2B2x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2=2 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29%2F%28x%5E2%2B2x%2B1%29=2 ALIGN=MIDDLE  ALT=\"%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29%2F%28x%5E2%2B2x%2B1%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29=2%28x%5E2%2B2x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29=2%28x%5E2%2B2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29=2x%5E2%2B4x%2B2 ALIGN=MIDDLE  ALT=\"%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29=2x%5E2%2B4x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29-%282x%5E2%2B4x%2B2%29=0 ALIGN=MIDDLE  ALT=\"%28x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1%29-%282x%5E2%2B4x%2B2%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=x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1-2x%5E2-4x-2=0 ALIGN=MIDDLE  ALT=\"x%5E5%2B4x%5E4%2B9x%5E3%2B15x%5E2%2B2x%2B1-2x%5E2-4x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5%2B4x%5E4%2B9x%5E3%2B13x%5E2-2x-1=0 ALIGN=MIDDLE  ALT=\"x%5E5%2B4x%5E4%2B9x%5E3%2B13x%5E2-2x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The rational number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=m%2Fn ALIGN=MIDDLE  ALT=\"x=m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > are integers with no common factors,<BR>\n" );
document.write( "that could be a solution to the equation above<BR>\n" );
document.write( "has an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > that is a factor of the independent term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is a factor of the leading coefficient, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F1=%28-1%29%2F%28-1%29 ALIGN=MIDDLE  ALT=\"x=1%2F1=%28-1%29%2F%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-1%29%2F1=1%2F%28-1%29 ALIGN=MIDDLE  ALT=\"x=%28-1%29%2F1=1%2F%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The problem with that, is that<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5%2B4x%5E4%2B9x%5E3%2B13x%5E2-2x-1=1%2B4%2B9%2B13-2-1=24 ALIGN=MIDDLE  ALT=\"x%5E5%2B4x%5E4%2B9x%5E3%2B13x%5E2-2x-1=1%2B4%2B9%2B13-2-1=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes the denominator zero.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is undefined.<BR>\n" );
document.write( "My conclusion is that there is no rational value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that will make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E3+-%28x-1%29%5E3+%2F+%28x%2B1%29%5E2+-+%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= 2 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );