document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1020275>Question 1020275</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	Please show work<BR>\n" );
document.write( "Consider the function <BR>\n" );
document.write( "f(x)=x^5-x^3<BR>\n" );
document.write( "	Determine the behaviors as x&#8594;&#8734;  and as x&#8594;-&#8734;<BR>\n" );
document.write( "	Find all zeros. What is the multiplicity of roots (zeros)?<BR>\n" );
document.write( "	When will the function cross the x-axis? <BR>\n" );
document.write( "        When will it cross the y-axis?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #636184 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=636184\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's do some modification.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E5-x%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E5-x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So as x get large in the positive direction, the right hand side is positive and the left hand term is positive so,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3Einfinity%2Cf%28x%29=infinity%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3Einfinity%2Cf%28x%29=infinity%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "SImilarly, as x get large in the negative direction, the right hand side is still positive and the left hand term turns negative so,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3Einfinity%2Cf%28x%29=-infinity%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3Einfinity%2Cf%28x%29=-infinity%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factor even more,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%28x%5E2-1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%28x%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%28x%2B1%29%28x-1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%28x%2B1%29%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So there are zeros at <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=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<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;\" >.<BR>\n" );
document.write( "The root at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has multiplicity of 3.<BR>\n" );
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document.write( "Odd multiplicity on the zero root means it crosses three times.<BR>\n" );
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document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(0,0)<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=cf14.JPG\">. </SPAN>\n" );
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