document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=999619>Question 999619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the end behavior of the graph of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #617224 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=617224\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  What is the end behavior of the graph of the polynomial function?\r<BR>\n" );
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document.write( "recall:\r<BR>\n" );
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document.write( "All even-degree polynomials behave, on their ends, like quadratics, and all odd-degree polynomials behave, on their ends, like cubics.\r<BR>\n" );
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document.write( "consider are the sign and the degree of the leading term; <BR>\n" );
document.write( "in your case the exponent says that this is a degree 6 polynomial, so the graph will behave roughly like a quadratic: up on both ends or down on both ends<BR>\n" );
document.write( "since the sign on the leading coefficient is positive, the graph will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=up ALIGN=MIDDLE  ALT=\"up\"   DISABLED_style= \"max-width:90%; height:auto;\" > on<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+both ALIGN=MIDDLE  ALT=\"+both\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ends ALIGN=MIDDLE  ALT=\"ends\"   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=+f%28x%29+=+3x%5E6+%2B+30x%5E5+%2B+75x%5E4 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E6+%2B+30x%5E5+%2B+75x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=>even-degree polynomial with positive leading coefficient which means the graph will behave roughly like a quadratic an will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=up ALIGN=MIDDLE  ALT=\"up\"   DISABLED_style= \"max-width:90%; height:auto;\" > on<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+both ALIGN=MIDDLE  ALT=\"+both\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ends ALIGN=MIDDLE  ALT=\"ends\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "now we can find solutions to this function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+3x%5E4%28x%5E2+%2B+10x+%2B+25%29 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E4%28x%5E2+%2B+10x+%2B+25%29\"   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=+f%28x%29+=+3x%5E4%28x%5E2+%2B+5x%2B5x+%2B+25%29 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E4%28x%5E2+%2B+5x%2B5x+%2B+25%29\"   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=+f%28x%29+=+3x%5E4%28%28x%5E2+%2B+5x%29%2B%285x+%2B+25%29%29 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E4%28%28x%5E2+%2B+5x%29%2B%285x+%2B+25%29%29\"   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=+f%28x%29+=+3x%5E4%28x%28x+%2B+5%29%2B5%28x+%2B+5%29%29 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E4%28x%28x+%2B+5%29%2B5%28x+%2B+5%29%29\"   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=+f%28x%29+=+3x%5E4%28x+%2B+5%29%5E2 ALIGN=MIDDLE  ALT=\"+f%28x%29+=+3x%5E4%28x+%2B+5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "roots:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+3x%5E4%28x+%2B+5%29%5E2 ALIGN=MIDDLE  ALT=\"+0+=+3x%5E4%28x+%2B+5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+3x%5E4 ALIGN=MIDDLE  ALT=\"+0+=+3x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (multiplicity four)<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+%28x+%2B+5%29%5E2 ALIGN=MIDDLE  ALT=\"+0+=+%28x+%2B+5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (multiplicity two)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-20%2C+1500%2C+3x%5E6+%2B+30x%5E5+%2B+75x%5E4%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-20%2C+1500%2C+3x%5E6+%2B+30x%5E5+%2B+75x%5E4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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