document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197185>Question 1197185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Analyze and sketch the polynomial functions and complete the charts below. State the degree and <BR>\n" );
document.write( "sign of the leading coefficient of the polynomial functions.  Determine the end behavior of the <BR>\n" );
document.write( "graph of the functions.  For 5b, write the polynomial function as a product of linear factors (in <BR>\n" );
document.write( "factored form). <BR>\n" );
document.write( "     a.  f(x) =  -  7(x + 2)3(x  -  1)2(x  -  3)   Zeros   Multiplicity   Crosses/Touches<BR>\n" );
document.write( "       Degree<BR>\n" );
document.write( "        Sign<BR>\n" );
document.write( "        End behavior<BR>\n" );
document.write( "        <BR>\n" );
document.write( "      b.  m(x) = x3  -  4x2                             <BR>\n" );
document.write( "      Factored form:  m(x) = <BR>\n" );
document.write( "      Degree            Zeros   Multiplicity   Crosses/Touches<BR>\n" );
document.write( "       Sign<BR>\n" );
document.write( "       End behavior </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830342 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=830342\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-+7%28x+%2B+2%293%28x+-+1%292%28x+-+3%29+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+-+7%28x+%2B+2%293%28x+-+1%292%28x+-+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "zeros/multiplicity /Crosses/Touches:  \r<BR>\n" );
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document.write( "zeros:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+-+7%28x+%2B+2%293%28x+-+1%292%28x+-+3%29+%7D%7D+%0D%0A%7B%7B%7B0=+%28x+%2B+2%29+%7D%7D+=%3E%7B%7B%7Bx=-2 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=0=+%28x+-1%29+%7D%7D+=%3E%7B%7B%7Bx=1 ALIGN=MIDDLE  ALT=\"0=+%28x+-1%29+%7D%7D+=%3E%7B%7B%7Bx=1\"   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=0=+%28x+-3%29+%7D%7D+=%3E%7B%7B%7Bx=3 ALIGN=MIDDLE  ALT=\"0=+%28x+-3%29+%7D%7D+=%3E%7B%7B%7Bx=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "multiplicity:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (Multiplicity of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   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=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (Multiplicity of <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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (Multiplicity of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Crosses/Touches: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Crosses\r<BR>\n" );
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document.write( "If a root of a polynomial has odd multiplicity, the graph will cross the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >-axis at the the root. \r<BR>\n" );
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document.write( "Degree: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   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+=+-42x%5E3+%2B+84x%5E2+%2B+210x+-+252 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-42x%5E3+%2B+84x%5E2+%2B+210x+-+252\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Sign: There is two sign change, so there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > positive root\r<BR>\n" );
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document.write( "End behavior:  \r<BR>\n" );
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document.write( "as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-%3E-infinity ALIGN=MIDDLE  ALT=\"x+-%3E-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-%3E+infinity ALIGN=MIDDLE  ALT=\"f%28x%29-%3E+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-%3Einfinity ALIGN=MIDDLE  ALT=\"x+-%3Einfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-%3E+-infinity ALIGN=MIDDLE  ALT=\"f%28x%29-%3E+-infinity\"   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=+graph%28+600%2C+600%2C+-10%2C+10%2C+-350%2C+350%2C+-42x%5E3+%2B+84x%5E2+%2B+210x+-+252%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-350%2C+350%2C+-42x%5E3+%2B+84x%5E2+%2B+210x+-+252%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28x%29+=+x%5E3+-+4x%5E2 ALIGN=MIDDLE  ALT=\"m%28x%29+=+x%5E3+-+4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Factored form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28x%29+=x%5E2%28x+-+4%29 ALIGN=MIDDLE  ALT=\"m%28x%29+=x%5E2%28x+-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Degree Zeros Multiplicity Crosses/Touches:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x%5E2 ALIGN=MIDDLE  ALT=\"0=x%5E2\"   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;\" >(Multiplicity of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x-4 ALIGN=MIDDLE  ALT=\"0=x-4\"   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 of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Sign: no sign changes, so there are no negative roots\r<BR>\n" );
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document.write( "End behavior:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-%3Einfinity ALIGN=MIDDLE  ALT=\"x+-%3Einfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-%3E+infinity ALIGN=MIDDLE  ALT=\"f%28x%29-%3E+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-%3E-infinity ALIGN=MIDDLE  ALT=\"x+-%3E-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-%3E+-infinity ALIGN=MIDDLE  ALT=\"f%28x%29-%3E+-infinity\"   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=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%28x+-+4%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%28x+-+4%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );