document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1034396>Question 1034396</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Sketch the graph of the polynomial function f(x) = &#8210;x4 + 5x3 + 9x2 &#8210; 45x in the interval [&#8210;4, 6].\r<BR>\n" );
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document.write( "     1) Write the polynomial function in factored form.<BR>\n" );
document.write( "     2) Sketch the graph of the polynomial function, labeling key points on the graph.<BR>\n" );
document.write( "     3) Describe the characteristics of the graph including: zeros, y-intercept, relative maximum and minimum and end behavior.<BR>\n" );
document.write( "    <BR>\n" );
document.write( "(3 points)\r<BR>\n" );
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document.write( "    \r<BR>\n" );
document.write( "\n" );
document.write( "2. Sketch the graph of the polynomial function f(x) = &#8210;x3 &#8210; 4x2 + 4x + 16 in the interval [&#8210;5, 3].\r<BR>\n" );
document.write( "\n" );
document.write( "     1) Write the polynomial function in factored form.<BR>\n" );
document.write( "     2) Sketch the graph of the polynomial function, labeling key points on the graph.<BR>\n" );
document.write( "     3) Describe the characteristics of the graph including: zeros, y-intercept, relative maximum and minimum and end behavior.<BR>\n" );
document.write( "    <BR>\n" );
document.write( "(3 points)<BR>\n" );
document.write( ".<BR>\n" );
document.write( " \r<BR>\n" );
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document.write( "3. Use what you have learned about the Remainder Theorem to show that X +1  is a factor of  19x42 + 18x -1.\r<BR>\n" );
document.write( "\n" );
document.write( "(3 points)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #649206 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=649206\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+-4+%2C+6+%5D%22 ALIGN=MIDDLE  ALT=\"%22%5B+-4+%2C+6+%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "1) Notice that two \"parts\" of the polynomial share <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29 ALIGN=MIDDLE  ALT=\"%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a common factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E4+%2B+5x%5E3=-x%5E3%28x-5%29 ALIGN=MIDDLE  ALT=\"-x%5E4+%2B+5x%5E3=-x%5E3%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-45x=9x%28x-5%29 ALIGN=MIDDLE  ALT=\"9x%5E2-45x=9x%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then factor by parts.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x=-x%5E3%28x-5%29%2B9x%28x-5%29=%28-x%5E3%2B9x%29%28x-5%29=-x%28x%5E2-9%29%28x-5%29=highlight%28-x%28x%2B3%29%28x-3%29%28x-5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Alternatively, youu could take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out as a common factor for the whole polynomial first:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x=-x%28x%5E3-5x%5E2-9x%5E2%2B45%29=-x%28x%5E2%28x-5%29-9%28x-5%29%29=-x%28x%5E2-9%29%28x-5%29=highlight%28-x%28x%2B3%29%28x-3%29%28x-5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "2) and 3) Each teacher/ instructor may have slightly different expectations about how to \"sketch the graph of the polynomial function, labeling key points on the graph\", but the calculations needed for the sketch would involve finding the values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the zeros, minima, and maxima, and figuring out the end behavior of the function.<BR>\n" );
document.write( "The y-intercept is easy:<BR>\n" );
document.write( "For <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;\" > , the value of the function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=-0%5E4+%2B+5%2A0%5E3+%2B+9%2A0%5E2-45%2A0=highlight%280%29 ALIGN=MIDDLE  ALT=\"f%280%29=-0%5E4+%2B+5%2A0%5E3+%2B+9%2A0%5E2-45%2A0=highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we can say that the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We know that the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+0+%2C+0+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+0+%2C+0+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the origin, is part of the graph of the function, and that will be the first point we plot in our sketch.<BR>\n" );
document.write( "The \"end behavior\" we are concerned with for the sketch in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+-4+%2C+6+%5D%22 ALIGN=MIDDLE  ALT=\"%22%5B+-4+%2C+6+%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is what happens at <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%28x%5E2-9%29%28x-5%29 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%28x%5E2-9%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%29=-%28-4%29%28%28-4%29%5E2-9%29%28%28-4%29-5%29=4%2816-9%29%28-9%29=4%2A7%2A%28-9%29=highlight%28-252%29 ALIGN=MIDDLE   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%286%29=-6%286%5E2-9%29%286-5%29=-6%2836-9%29%2A1=-6%2A27=highlight%28-162%29 ALIGN=MIDDLE  ALT=\"f%286%29=-6%286%5E2-9%29%286-5%29=-6%2836-9%29%2A1=-6%2A27=highlight%28-162%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In general, the end behavior of a function involves limits,<BR>\n" );
document.write( "but in the case of polynomials,<BR>\n" );
document.write( "we learned early in algebra that the end behavior is determined by<BR>\n" );
document.write( "the leading coefficient (whether it is positive or negative), and<BR>\n" );
document.write( "the degree of the polynomial (whether it is even or odd).<BR>\n" );
document.write( "A polynomial's end behavior is the same as the end behavior of its first term,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x\"   DISABLED_style= \"max-width:90%; height:auto;\" > behaves as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E4 ALIGN=MIDDLE  ALT=\"-x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the ends of the graph: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3Einfinity%2Cf%28x%29%29=-infinity ALIGN=MIDDLE  ALT=\"lim%28x-%3Einfinity%2Cf%28x%29%29=-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(The polynomial is negative at both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=infinity ALIGN=MIDDLE  ALT=\"x=infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > ends, and<BR>\n" );
document.write( "as we go towards those ends, the polynomial's absolute value increases without  bounds).<BR>\n" );
document.write( "The zeros are obvious from the factored form of the polynomial, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%28x%2B3%29%28x-3%29%28x-5%29 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%28x%2B3%29%28x-3%29%28x-5%29\"   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=0 ALIGN=MIDDLE  ALT=\"f%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that make it factor zero,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%280%29 ALIGN=MIDDLE  ALT=\"x=highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%28-3%29 ALIGN=MIDDLE  ALT=\"x=highlight%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%283%29 ALIGN=MIDDLE  ALT=\"x=highlight%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%285%29 ALIGN=MIDDLE  ALT=\"x=highlight%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At each one of those zeros, the factor involved changes sign, and so does the function, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%3C0 ALIGN=MIDDLE  ALT=\"f%28x%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%3C=x%3C-3 ALIGN=MIDDLE  ALT=\"-4%3C=x%3C-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%3E0 ALIGN=MIDDLE  ALT=\"f%28x%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%3Cx%3C0 ALIGN=MIDDLE  ALT=\"-3%3Cx%3C0\"   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%3C0 ALIGN=MIDDLE  ALT=\"f%28x%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C3 ALIGN=MIDDLE  ALT=\"0%3Cx%3C3\"   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%3E0 ALIGN=MIDDLE  ALT=\"f%28x%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3Cx%3C5 ALIGN=MIDDLE  ALT=\"3%3Cx%3C5\"   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%3C0 ALIGN=MIDDLE  ALT=\"f%28x%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%3C=x%3C=6 ALIGN=MIDDLE  ALT=\"5%3C=x%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is useful information for your sketch,<BR>\n" );
document.write( "and would lead you to conclude that the polynomial must have<BR>\n" );
document.write( "a relative maximum in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-3+%2C+0+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-3+%2C+0+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%3Cx%3C0 ALIGN=MIDDLE  ALT=\"-3%3Cx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "a relative minimum in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+0+%2C+3+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+0+%2C+3+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C3 ALIGN=MIDDLE  ALT=\"0%3Cx%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "a relative maximum in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+3+%2C+5+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+3+%2C+5+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3Cx%3C5 ALIGN=MIDDLE  ALT=\"3%3Cx%3C5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "To locate those relative maxima and minimum,<BR>\n" );
document.write( "we need the derivative, and we need to calculate the zeros of the derivative.<BR>\n" );
document.write( "The derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-4x%5E3%2B5%283x%5E2%29%2B9%282x%29-45=-4x%5E3%2B15x%5E2%2B18x-45 ALIGN=MIDDLE  ALT=\"df%2Fdx=-4x%5E3%2B5%283x%5E2%29%2B9%282x%29-45=-4x%5E3%2B15x%5E2%2B18x-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That polynomial cannot be factored, but it must change signs at least once,<BR>\n" );
document.write( "because its degree is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is an odd number.<BR>\n" );
document.write( "In fact, it is not hard to see that it changes signs <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times, because<BR>\n" );
document.write( "for <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;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-4%28-10%29%5E3%2B15%28-10%29%5E2%2B18%28-10%29-45=4000%2B1500-180-45%3E0 ALIGN=MIDDLE  ALT=\"df%2Fdx=-4%28-10%29%5E3%2B15%28-10%29%5E2%2B18%28-10%29-45=4000%2B1500-180-45%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "for <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;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-45%3C0 ALIGN=MIDDLE  ALT=\"df%2Fdx=-45%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "for <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;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-4%2A2%5E3%2B15%2A2%5E2%2B18%2A2-45=-32%2B60%2B36-45=19%3E0 ALIGN=MIDDLE  ALT=\"df%2Fdx=-4%2A2%5E3%2B15%2A2%5E2%2B18%2A2-45=-32%2B60%2B36-45=19%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "for <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;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-4%2A10%5E3%2B15%2A10%5E2%2B18%2A10-45=-4000%2B1500-180-45=-250-180-45%3C0 ALIGN=MIDDLE  ALT=\"df%2Fdx=-4%2A10%5E3%2B15%2A10%5E2%2B18%2A10-45=-4000%2B1500-180-45=-250-180-45%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So it must have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > zeros, but those zeros must be irrational numbers,<BR>\n" );
document.write( "because we cannot factor it.<BR>\n" );
document.write( "We cannot find any rational zeros of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the independent term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-45 ALIGN=MIDDLE  ALT=\"-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "All we can do is use a calculator to find that the zeros of derivative.<BR>\n" );
document.write( "A graphing calculator (if we know how to use it) would make it easier,<BR>\n" );
document.write( "but with any calculator (or a spreadsheet) we can try values to \"zero in\" on those zeros,<BR>\n" );
document.write( "and find that the zeros of derivative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-4x%5E3%2B15x%5E2%2B18x-45 ALIGN=MIDDLE  ALT=\"df%2Fdx=-4x%5E3%2B15x%5E2%2B18x-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > are approximately<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.87 ALIGN=MIDDLE  ALT=\"x=-1.87\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.44 ALIGN=MIDDLE  ALT=\"x=1.44\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4.18 ALIGN=MIDDLE  ALT=\"x=4.18\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can use the calculator to find the value of <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;\" > at those points as approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70.70 ALIGN=MIDDLE  ALT=\"70.70\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-35.51 ALIGN=MIDDLE  ALT=\"-35.51\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29.04 ALIGN=MIDDLE  ALT=\"29.04\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells you to include the points<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-1.87+%2C+70.70+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-1.87+%2C+70.70+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+1.44+%2C+-35.51+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+1.44+%2C+-35.51+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+4.18+%2C+29.04+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+4.18+%2C+29.04+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the sketch.<BR>\n" );
document.write( "So far, you have 8 points of your sketch:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C7%2C-350%2C150%2C+grid%280%29%2C%0D%0Acircle%28-4%2C-252%2C0.1%29%2Ccircle%286%2C-162%2C0.1%29%2C%0D%0Acircle%28-3%2C0%2C0.1%29%2Ccircle%283%2C0%2C0.1%29%2C%0D%0Acircle%280%2C0%2C0.1%29%2Ccircle%285%2C0%2C0.1%29%2C%0D%0Acircle%28-1.87%2C70.7%2C0.1%29%2Ccircle%281.44%2C-35.51%2C0.1%29%2Ccircle%284.18%2C29.04%2C0.1%29%2C%0D%0Acircle%28-4%2C-252%2C0.2%29%2Ccircle%286%2C-162%2C0.2%29%2C%0D%0Acircle%28-3%2C0%2C0.2%29%2Ccircle%283%2C0%2C0.2%29%2C%0D%0Acircle%280%2C0%2C0.2%29%2Ccircle%285%2C0%2C0.2%29%2C%0D%0Acircle%28-1.87%2C70.7%2C0.2%29%2Ccircle%281.44%2C-35.51%2C0.2%29%2Ccircle%284.18%2C29.04%2C0.2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For a sketch, all you have to do is join those points with a smooth curve, like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C7%2C-350%2C150%2C+grid%280%29%2C%0D%0Agraph%28300%2C300%2C-5%2C7%2C-350%2C150%2C200%2C%28-x%5E4+%2B+5x%5E3+%2B+9x%5E2-45x%29sqrt%2824%2B2x-x%5E2%29%2Fsqrt%2824%2B2x-x%5E2%29%29%2C%0D%0Acircle%28-4%2C-252%2C0.1%29%2Ccircle%286%2C-162%2C0.1%29%2C%0D%0Acircle%28-3%2C0%2C0.1%29%2Ccircle%283%2C0%2C0.1%29%2C%0D%0Acircle%280%2C0%2C0.1%29%2Ccircle%285%2C0%2C0.1%29%2C%0D%0Acircle%28-1.87%2C70.7%2C0.1%29%2Ccircle%281.44%2C-35.51%2C0.1%29%2Ccircle%284.18%2C29.04%2C0.1%29%2C%0D%0Acircle%28-4%2C-252%2C0.2%29%2Ccircle%286%2C-162%2C0.2%29%2C%0D%0Acircle%28-3%2C0%2C0.2%29%2Ccircle%283%2C0%2C0.2%29%2C%0D%0Acircle%280%2C0%2C0.2%29%2Ccircle%285%2C0%2C0.2%29%2C%0D%0Acircle%28-1.87%2C70.7%2C0.2%29%2Ccircle%281.44%2C-35.51%2C0.2%29%2Ccircle%284.18%2C29.04%2C0.2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E3-4x%5E2%2B4x%2B16 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E3-4x%5E2%2B4x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+-5+%2C+3+%5D%22 ALIGN=MIDDLE  ALT=\"%22%5B+-5+%2C+3+%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "1) The rational zero theorem tells us that a rational zero of that polynomial must be of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the leading coefficient <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<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the independent term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The choices are -16, -8, -4, -4, -2, -1, 1, 2, 4, 8, and 16.<BR>\n" );
document.write( "Trying them , starting with the easy ones, we find that <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;\" > 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;\" > are not zeros.<BR>\n" );
document.write( "However, <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;\" > is a zero, and dividing the polynomial by <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;\" ><BR>\n" );
document.write( "we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-x%5E2%2B6x%2B8 ALIGN=MIDDLE  ALT=\"%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-x%5E2%2B6x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "whose zeros are <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;\" > , 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;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-x%5E2%2B6x%2B8=-%28x%2B4%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-x%5E2%2B6x%2B8=-%28x%2B4%29%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-%28x%2B4%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28-x%5E3-4x%5E2%2B4x%2B16%29%2F%28x-2%29=-%28x%2B4%29%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E3-4x%5E2%2B4x%2B16%29=highlight%28-%28x-2%29%28x%2B4%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E3-4x%5E2%2B4x%2B16%29=highlight%28-%28x-2%29%28x%2B4%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) and 3) Two points we need to include in the sketch are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-5+%2C+21+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-5+%2C+21+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-5%29=-%28-5-2%29%28-5%2B4%29%28-5%2B2%29=-%28-7%29%28-1%29%28-3%29=21 ALIGN=MIDDLE  ALT=\"f%28-5%29=-%28-5-2%29%28-5%2B4%29%28-5%2B2%29=-%28-7%29%28-1%29%28-3%29=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+3+%2C+-35+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+3+%2C+-35+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29=-%283-2%29%283%2B4%29%283%2B2%29=-%281%29%287%29%285%29-35 ALIGN=MIDDLE  ALT=\"f%283%29=-%283-2%29%283%2B4%29%283%2B2%29=-%281%29%287%29%285%29-35\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Other points for the sketch are zeros, y-intercept, minimum and maximum.<BR>\n" );
document.write( "The zeros of the function are <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;\" > , <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;\" > , 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;\" > .<BR>\n" );
document.write( "Those zeros mark points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-4+%2C+0+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-4+%2C+0+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-2+%2C+0+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-2+%2C+0+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+2+%2C+0+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+2+%2C+0+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "also to be included in the sketch of the graph.<BR>\n" );
document.write( "The y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=-0%5E3-4%2A0%5E2%2B4%2A0%2B16=highlight%2816%29 ALIGN=MIDDLE  ALT=\"f%280%29=-0%5E3-4%2A0%5E2%2B4%2A0%2B16=highlight%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which tells you that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+0+%2C+16+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+0+%2C+16+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one point on the graph.<BR>\n" );
document.write( "The maxima and minima are zeros of derivative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=-3x%5E2-4%282x%29%2B4=-3x%5E2-8x%2B4 ALIGN=MIDDLE  ALT=\"df%2Fdx=-3x%5E2-4%282x%29%2B4=-3x%5E2-8x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those zeros are the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2-8x%2B4=0 ALIGN=MIDDLE  ALT=\"-3x%5E2-8x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-4+%2B-+sqrt%2828%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28-4+%2B-+sqrt%2828%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3.10 ALIGN=MIDDLE  ALT=\"x=-3.10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.43 ALIGN=MIDDLE  ALT=\"x=0.43\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The minimum and maximum of the function are<BR>\n" );
document.write( "approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3.10%29=about-5.05 ALIGN=MIDDLE  ALT=\"f%28-3.10%29=about-5.05\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280.43%29=about16.9 ALIGN=MIDDLE  ALT=\"f%280.43%29=about16.9\"   DISABLED_style= \"max-width:90%; height:auto;\" > respectively.<BR>\n" );
document.write( "So, points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+-3.10+%2C+-5.05+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+-3.10+%2C+-5.05+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+0.43+%2C+16.9+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+0.43+%2C+16.9+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "also need to be included in the sketch of the graph.<BR>\n" );
document.write( "The sketch would look something like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-6%2C4%2C-50%2C50%2C%0D%0Agraph%28300%2C300%2C-6%2C4%2C-50%2C50%2C200%2C%28-x%5E3-4x%5E2%2B4x%2B16%29sqrt%2815-2x-x%5E2%29%2Fsqrt%2815-2x-x%5E2%29%29%2C%0D%0Acircle%28-5%2C21%2C0.1%29%2Ccircle%283%2C-35%2C0.1%29%2C%0D%0Acircle%28-4%2C0%2C0.1%29%2Ccircle%28-2%2C0%2C0.1%29%2C%0D%0Acircle%282%2C0%2C0.1%29%2Ccircle%280%2C16%2C0.1%29%2C%0D%0Acircle%28-3.1%2C-5.05%2C0.1%29%2Ccircle%280.43%2C16.9%2C0.1%29%2C%0D%0Acircle%28-5%2C21%2C0.2%29%2Ccircle%283%2C-35%2C0.2%29%2C%0D%0Acircle%28-4%2C0%2C0.2%29%2Ccircle%28-2%2C0%2C0.2%29%2C%0D%0Acircle%282%2C0%2C0.2%29%2Ccircle%280%2C16%2C0.2%29%2C%0D%0Acircle%28-3.1%2C-5.05%2C0.2%29%2Ccircle%280.43%2C16.9%2C0.2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "3) According to the Remainder Theorem,<BR>\n" );
document.write( "the remainder of the division of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=19x%5E42%2B18x-1 ALIGN=MIDDLE  ALT=\"f%28x%29=19x%5E42%2B18x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29=%28x-%28-1%29%29 ALIGN=MIDDLE  ALT=\"%28x%2B1%29=%28x-%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29 ALIGN=MIDDLE  ALT=\"f%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since that remainder is zero:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29=19%28-1%29%5E42%2B18%28-1%29-1=19%281%29-18-1=19-18-1=0 ALIGN=MIDDLE  ALT=\"f%28-1%29=19%28-1%29%5E42%2B18%28-1%29-1=19%281%29-18-1=19-18-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that means that <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;\" > is divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or in other words that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=19x%5E42%2B18x-1 ALIGN=MIDDLE  ALT=\"f%28x%29=19x%5E42%2B18x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );