document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189459>Question 1189459</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the polynomial Function : F(x)=-2(x + 1/2) (x+4)^3<BR>\n" );
document.write( "a)List each real zero and its multiplicity:<BR>\n" );
document.write( "b)Determine whether graph crosses or touches the x-axis at each x-intercept:<BR>\n" );
document.write( "c)Determine the behavior of the graph near each x-intercept(zero):<BR>\n" );
document.write( "d)Determine the maximum number of turning point on the graph:<BR>\n" );
document.write( "e)Determine the end behavior, that is finding the power function that the graph of<BR>\n" );
document.write( "f resembles for large values of |x|:<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 #820848 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=820848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Graph this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-50%2C25%2C2%28x%2B0.5%29%28x%2B4%29%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-50%2C25%2C2%28x%2B0.5%29%28x%2B4%29%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C2%28x%2B0.5%29%28x%2B4%29%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C2%28x%2B0.5%29%28x%2B4%29%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By inspection, x=-4 is a root with multiplicity 3, and so is x=-0.5. Both of those make the parentheses 0, and they cross the x-axis.<BR>\n" );
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document.write( "Take the derivative of the function:<BR>\n" );
document.write( "f'(x)=2(x+0.5)*3(x+4)^2+(x+4)^3(2)<BR>\n" );
document.write( "as x approaches -4 from the either side, the derivative is negative, or the slope negative. <BR>\n" );
document.write( "As x approaches 0.5 from either side, the slope is positive.<BR>\n" );
document.write( "There is one turning point where the derivative is 0.<BR>\n" );
document.write( "2(x+0.5)*3(x+4)^2+2(x+4)^3=0<BR>\n" );
document.write( "divide both sides by (x+4)^2<BR>\n" );
document.write( "6x+3+2(x+4)=0<BR>\n" );
document.write( "8x+11=0<BR>\n" );
document.write( "x=-1.375<BR>\n" );
document.write( "y=-31.65<BR>\n" );
document.write( "-<BR>\n" );
document.write( "end behavior negative is (-x)^4 positive oo<BR>\n" );
document.write( "and for positive is x^4 positive oo<BR>\n" );
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