document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=953542>Question 953542</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is a fifth-degree polynomial with <BR>\n" );
document.write( "the x-intercepts -4, 1, and 7, <BR>\n" );
document.write( "y-intercept 112 and <BR>\n" );
document.write( "f (x) &#8805; 0 for x &#8804; 7\r<BR>\n" );
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document.write( "I know it would have to have (x+4)(x-1)(x+7) and possibly either squared or cubics for exponents but I am confused about how I would insert the 112 and the f (x) &#8805; 0 for x &#8804; 7. I don't understand what those last two are asking me to do to make it true.\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #582349 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=582349\">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=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   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=112 ALIGN=MIDDLE  ALT=\"112\"   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=+7+ ALIGN=MIDDLE  ALT=\"x%3C=+7+\"   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+-%28-4%29%29 ALIGN=MIDDLE  ALT=\"%28x+-%28-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+1%29 ALIGN=MIDDLE  ALT=\"%28x+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-7%29 ALIGN=MIDDLE  ALT=\"+%28x-7%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+4%29%28x+-1%29%28x-7%29+%2Ag%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x+%2B+4%29%28x+-1%29%28x-7%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=+7 ALIGN=MIDDLE  ALT=\"x+%3C=+7\"   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+4%29%28x+-1%29%28x-7%29+%2Ag%28x%29%3E=+0 ALIGN=MIDDLE  ALT=\"+%28x+%2B+4%29%28x+-1%29%28x-7%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=+7 ALIGN=MIDDLE  ALT=\"x+%3C=+7\"   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=+7 ALIGN=MIDDLE  ALT=\"x+%3C=+7\"   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+4 ALIGN=MIDDLE  ALT=\"x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > change signs when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+7 ALIGN=MIDDLE  ALT=\"x+%3C=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-7 ALIGN=MIDDLE  ALT=\"x+-7\"   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+4%29%28x+-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+A%28x+%2B+4%29%28x+-1%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+4%29%5E2+%28x-+1%29%5E2+%28x+-7%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+A%28x+%2B+4%29%5E2+%28x-+1%29%5E2+%28x+-7%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+4%29%5E2+%28x+-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x+%2B+4%29%5E2+%28x+-1%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=7 ALIGN=MIDDLE  ALT=\"x+%3C=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > reduces to needing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x+-7%29+%3E=0 ALIGN=MIDDLE  ALT=\"A%28x+-7%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=7 ALIGN=MIDDLE  ALT=\"x%3C=7\"   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+=+112 ALIGN=MIDDLE  ALT=\"f%280%29+=+112\"   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+4%5E2+%28-1%29%5E2+%28-7%29+=+112 ALIGN=MIDDLE  ALT=\"A+%2A+4%5E2+%28-1%29%5E2+%28-7%29+=+112\"   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+16+%281%29+%28-7%29+=+112 ALIGN=MIDDLE  ALT=\"A+%2A+16+%281%29+%28-7%29+=+112\"   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+%28-112%29+=+112 ALIGN=MIDDLE  ALT=\"A+%2A+%28-112%29+=+112\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A++=+112%2F-112 ALIGN=MIDDLE  ALT=\"A++=+112%2F-112\"   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 ALIGN=MIDDLE  ALT=\"+A+=+-1\"   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%29%28x+%2B+4%29%5E2+%28x+-+1%29%5E2+%28x+-+7%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28-1%29%28x+%2B+4%29%5E2+%28x+-+1%29%5E2+%28x+-+7%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%2Bx%5E4%2B41x%5E3%2B31x%5E2-184x%2B112 ALIGN=MIDDLE  ALT=\"f%28x%29+=-x%5E5%2Bx%5E4%2B41x%5E3%2B31x%5E2-184x%2B112\"   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+-5%2C+20%2C+-5%2C+450%2C%28-1%29%28x+%2B+4%29%5E2%2A+%28x+-+1%29%5E2%2A+%28x+-+7%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );