document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=959245>Question 959245</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Having a lot of difficulty visualizing this don't know where to put the exponents could really use explanations:\r<BR>\n" );
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document.write( "The function f is a fifth-degree polynomial with \r<BR>\n" );
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document.write( "the x-intercepts -5, 2, and 10, <BR>\n" );
document.write( "y-intercept 500 and <BR>\n" );
document.write( "f (x) &#8805; 0 for x &#8804; 10. \r<BR>\n" );
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document.write( "Find f (x).\r<BR>\n" );
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document.write( "f (x) =\r<BR>\n" );
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document.write( "I don't understand what f(x)&#8805;0 for x&#8804;10 means, please explain that and how to solve for the rest.\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #586342 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=586342\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> given:<BR>\n" );
document.write( "a fifth-degree polynomial with<BR>\n" );
document.write( "the x-intercepts <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "y-intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500 ALIGN=MIDDLE  ALT=\"500\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+%3E=+0 ALIGN=MIDDLE  ALT=\"f%28x%29+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=+10+ ALIGN=MIDDLE  ALT=\"x%3C=+10+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "By hypothesis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has factors <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-%28-5%29%29 ALIGN=MIDDLE  ALT=\"%28x+-%28-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+2%29 ALIGN=MIDDLE  ALT=\"%28x+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-10%29 ALIGN=MIDDLE  ALT=\"+%28x-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the Factor Theorem.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x+%2B+5%29%28x+-2%29%28x-10%29+%2Ag%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x+%2B+5%29%28x+-2%29%28x-10%29+%2Ag%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some 2nd degree polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To determine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "We need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%3E=+0 ALIGN=MIDDLE  ALT=\"f%28x%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+10 ALIGN=MIDDLE  ALT=\"x+%3C=+10\"   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=+%28x+%2B+5%29%28x+-2%29%28x-10%29+%2Ag%28x%29%3E=+0 ALIGN=MIDDLE  ALT=\"+%28x+%2B+5%29%28x+-2%29%28x-10%29+%2Ag%28x%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+10 ALIGN=MIDDLE  ALT=\"x+%3C=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note that each factor is negative for some <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+10 ALIGN=MIDDLE  ALT=\"x+%3C=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "There a few ways to deal with this; here is one way.\r<BR>\n" );
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document.write( "Note that both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+5 ALIGN=MIDDLE  ALT=\"x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > change signs when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+10 ALIGN=MIDDLE  ALT=\"x+%3C=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-10 ALIGN=MIDDLE  ALT=\"x+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not.\r<BR>\n" );
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document.write( "Since we need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be quadratic, we can take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+A%28x+%2B+5%29%28x+-2%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+A%28x+%2B+5%29%28x+-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+A%28x+%2B+5%29%5E2+%28x-+2%29%5E2+%28x+-10%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+A%28x+%2B+5%29%5E2+%28x-+2%29%5E2+%28x+-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Note: that we have not changed the x-intercepts by repeating some of the factors \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+5%29%5E2+%28x+-2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x+%2B+5%29%5E2+%28x+-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is never negative, needing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%28x%29+%3E=+0 ALIGN=MIDDLE  ALT=\"f+%28x%29+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=10 ALIGN=MIDDLE  ALT=\"x+%3C=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > reduces to needing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x+-10%29+%3E=0 ALIGN=MIDDLE  ALT=\"A%28x+-10%29+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=10 ALIGN=MIDDLE  ALT=\"x%3C=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "*This is guaranteed if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%3C+0 ALIGN=MIDDLE  ALT=\"A+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Finally, we use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+500 ALIGN=MIDDLE  ALT=\"f%280%29+=+500\"   DISABLED_style= \"max-width:90%; height:auto;\" > to determine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2A+5%5E2+%28-2%29%5E2+%28-10%29+=+500 ALIGN=MIDDLE  ALT=\"A+%2A+5%5E2+%28-2%29%5E2+%28-10%29+=+500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2A+25+%2A4+%28-10%29+=+500 ALIGN=MIDDLE  ALT=\"A+%2A+25+%2A4+%28-10%29+=+500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2A+%28-1000%29+=+500 ALIGN=MIDDLE  ALT=\"A+%2A+%28-1000%29+=+500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A++=+500%2F-1000 ALIGN=MIDDLE  ALT=\"A++=+500%2F-1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+-1%2F2 ALIGN=MIDDLE  ALT=\"+A+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is indeed negative.\r<BR>\n" );
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document.write( "Hence, we can take \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28-1%2F2%29%28x+%2B+5%29%5E2+%28x+-+2%29%5E2+%28x+-+10%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28-1%2F2%29%28x+%2B+5%29%5E2+%28x+-+2%29%5E2+%28x+-+10%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+=%28-1%2F2%29%28x%5E5-4+x%5E4-71+x%5E3%2B50+x%5E2%2B700+x-1000%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=%28-1%2F2%29%28x%5E5-4+x%5E4-71+x%5E3%2B50+x%5E2%2B700+x-1000%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+=-x%5E5%2F2%2B2+x%5E4%2B%2871x%5E3%29%2F2-25x%5E2-350x%2B500 ALIGN=MIDDLE  ALT=\"f%28x%29+=-x%5E5%2F2%2B2+x%5E4%2B%2871x%5E3%29%2F2-25x%5E2-350x%2B500\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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