document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=976266>Question 976266</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please a need help with this<BR>\n" );
document.write( "QUESTION\r<BR>\n" );
document.write( "\n" );
document.write( "(a)prove that the equation mx(x^2+2x+3) = x^2-2x-3 has exactly one real root if m=1 and exactly 3 real roots if m=-2/3. \r<BR>\n" );
document.write( "\n" );
document.write( "(b) prove that tan t = sin2t/1+cos2t<BR>\n" );
document.write( "and hence obtain the value of tan 15 degrees and 45 degrees  and express the result in standard from\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "THANKS\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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 #597954 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=597954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE EASY ONE:<BR>\n" );
document.write( "(b) prove that tan t = sin 2t/(1+cos 2t) or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28t%29=sin%282t%29%2F%281%2Bcos%282t%29%29 ALIGN=MIDDLE  ALT=\"tan%28t%29=sin%282t%29%2F%281%2Bcos%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(Parentheses matter because   sin 2t/(1+cos 2t)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282t%29%2F%281%2Bcos%282t%29%29 ALIGN=MIDDLE  ALT=\"sin%282t%29%2F%281%2Bcos%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282t%29%2F1%2Bcos%282t%29 ALIGN=MIDDLE  ALT=\"sin%282t%29%2F1%2Bcos%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "sin 2t/1+cos 2t =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282t%29=2sin%28t%29cos%28t%29 ALIGN=MIDDLE  ALT=\"sin%282t%29=2sin%28t%29cos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29%2Bcos%5E2%28t%29-sin%5E2%28t%29 ALIGN=MIDDLE  ALT=\"cos%282t%29%2Bcos%5E2%28t%29-sin%5E2%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are popular trigonometric identities.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28t%29=sin%28t%29%2Fcos%28t%29 ALIGN=MIDDLE  ALT=\"tan%28t%29=sin%28t%29%2Fcos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28t%29%2Bsin%5E2%28t%29=1 ALIGN=MIDDLE  ALT=\"cos%5E2%28t%29%2Bsin%5E2%28t%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28t%29=1-sin%5E2%28t%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28t%29=1-sin%5E2%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > even better known.<BR>\n" );
document.write( "If we substitute, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282t%29%2F%281%2Bcos%282t%29%29 ALIGN=MIDDLE  ALT=\"sin%282t%29%2F%281%2Bcos%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28t%29cos%28t%29%2F%281%2Bcos%5E2%28t%29-sin%5E2%28t%29%29 ALIGN=MIDDLE  ALT=\"2sin%28t%29cos%28t%29%2F%281%2Bcos%5E2%28t%29-sin%5E2%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28t%29cos%28t%29%2F%28%281-sin%5E2%28t%29%29%2Bcos%5E2%28t%29%29 ALIGN=MIDDLE  ALT=\"2sin%28t%29cos%28t%29%2F%28%281-sin%5E2%28t%29%29%2Bcos%5E2%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28t%29cos%28t%29%2F%28cos%5E2%28t%29%2Bcos%5E2%28t%29%29 ALIGN=MIDDLE  ALT=\"2sin%28t%29cos%28t%29%2F%28cos%5E2%28t%29%2Bcos%5E2%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28t%29cos%28t%29%2F%282cos%5E2%28t%29%29 ALIGN=MIDDLE  ALT=\"2sin%28t%29cos%28t%29%2F%282cos%5E2%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28t%29%2Fcos%28t%29 ALIGN=MIDDLE  ALT=\"sin%28t%29%2Fcos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28t%29 ALIGN=MIDDLE  ALT=\"tan%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE OTHER PROBLEM:<BR>\n" );
document.write( "(a)prove that the equation mx(x^2+2x+3) = x^2-2x-3 has exactly one real root if m=1 and exactly 3 real roots if m=-2/3.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%28x%5E2%2B2x%2B3%29+=+x%5E2-2x-3 ALIGN=MIDDLE  ALT=\"mx%28x%5E2%2B2x%2B3%29+=+x%5E2-2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%28x%5E2%2B2x%2B3%29-x%5E2%2B2x%2B3=0 ALIGN=MIDDLE  ALT=\"mx%28x%5E2%2B2x%2B3%29-x%5E2%2B2x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3=0 ALIGN=MIDDLE  ALT=\"mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a cubic equation,<BR>\n" );
document.write( "and cubic equations are easy to solve only when the teacher designs them so that they will be easy to solve.<BR>\n" );
document.write( "The roots we look for are the zeros of the cubic polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In general, to know how many roots the cubic polynomial has, and to find their approximate values we can use calculus (or a graphing calculator).<BR>\n" );
document.write( " <BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-2%2F3 ALIGN=MIDDLE  ALT=\"m=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%282%28-2%2F3%29-1%29x%5E2%2B%283%28-2%2F3%29%2B2%29x%2B3=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the case of this cubic polynomial, we can actually find all three zeros easily,<BR>\n" );
document.write( "because with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%29=%28-2%2F3%29%2B%28-7%2F3%29%2B3=-9%2F3%2B3=-3%2B3=0 ALIGN=MIDDLE  ALT=\"P%281%29=%28-2%2F3%29%2B%28-7%2F3%29%2B3=-9%2F3%2B3=-3%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so <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;\" > is one of the zeros,<BR>\n" );
document.write( "so <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;\" > is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and dividing we find that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3=%28x-1%29%28%28-2%2F3%29x%5E2-3x-3%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3=%28x-1%29%28%28-2%2F3%29x%5E2-3x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the other two real zeros (if any) would be the solutions to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2F3%29x%5E2-3x-3=0 ALIGN=MIDDLE  ALT=\"%28-2%2F3%29x%5E2-3x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2-9x-9=0 ALIGN=MIDDLE  ALT=\"-2x%5E2-9x-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B9x%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B9x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3%29%28x%2B3%29=0 ALIGN=MIDDLE  ALT=\"%282x%2B3%29%28x%2B3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3%2F2 ALIGN=MIDDLE  ALT=\"x=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1 ALIGN=MIDDLE  ALT=\"m=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and I do not see an easy algebra way to figure out if that polynomial has 3 real zeros.<BR>\n" );
document.write( "There may be a simpler algebra way to prove it has only one,<BR>\n" );
document.write( "but I keep thinking calculus.<BR>\n" );
document.write( "That bothers me, because I suspect there is a simpler (and therefore better) solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "A cubic function, <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;\" > , may have a graph like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C1%2C5%2C1%2C5%2Cx%5E3-9x%5E2%2B27x-24%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C1%2C5%2C1%2C5%2Cx%5E3-9x%5E2%2B27x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C1%2C5%2C1%2C5%2C-x%5E3%2B9x%5E2-27x%2B30%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C1%2C5%2C1%2C5%2C-x%5E3%2B9x%5E2-27x%2B30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C5%2C15%2C5%2C95%2Cx%5E3-29x%5E2%2B285x-900%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C5%2C15%2C5%2C95%2Cx%5E3-29x%5E2%2B285x-900%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with no relative maximum or minimum,<BR>\n" );
document.write( "and then it would cross the x-axis (making <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;\" > ) just for only one value 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;\" > .<BR>\n" );
document.write( "Then, the equation <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;\" > would have exactly one real root.<BR>\n" );
document.write( "Cubic polynomials can also have graphs like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C3%2C9%2Cx%5E3-18x%5E2%2B107x-204%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C3%2C9%2Cx%5E3-18x%5E2%2B107x-204%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C3%2C9%2C-x%5E3%2B18x%5E2-107x%2B216%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C3%2C9%2C-x%5E3%2B18x%5E2-107x%2B216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with one maximum and one minimum,<BR>\n" );
document.write( "and in that case the graph may cross/touch the x-axis at one, two, or three points,<BR>\n" );
document.write( "as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C-3%2C1%2Cx%5E3-18x%5E2%2B107x-211%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C-3%2C1%2Cx%5E3-18x%5E2%2B107x-211%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C-3%2C1%2Cx%5E3-18x%5E2%2B107x-210.3849%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C-3%2C1%2Cx%5E3-18x%5E2%2B107x-210.3849%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C-2%2C2%2Cx%5E3-18x%5E2%2B107x-210%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C-2%2C2%2Cx%5E3-18x%5E2%2B107x-210%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In those cases, the equation <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;\" > would have respectively one real root, or two, or three.<BR>\n" );
document.write( "We would want to know if the maximum and minimum values 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;\" ><BR>\n" );
document.write( "are both positive, or both negative (one root), or<BR>\n" );
document.write( "one is zero (two roots), or<BR>\n" );
document.write( "one is positive and the other negative (3 roots).<BR>\n" );
document.write( "We can figure out if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=mx%28x%5E2%2B2x%2B3%29-x%5E2%2B2x%2B3=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=mx%28x%5E2%2B2x%2B3%29-x%5E2%2B2x%2B3=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a maximum and a minimum by looking at its derivative, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22P+%27+%28+x+%29%22 ALIGN=MIDDLE  ALT=\"%22P+%27+%28+x+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx ALIGN=MIDDLE  ALT=\"dP%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The derivative will be a quadratic function, and its real zeros (if any) indicate maximum and minimum.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=mx%5E3%2B%282m-1%29x%5E2%2B%283m%2B2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx=3mx%5E2%2B2%282m-1%29x%2B%283m%2B2%29 ALIGN=MIDDLE  ALT=\"dP%2Fdx=3mx%5E2%2B2%282m-1%29x%2B%283m%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1 ALIGN=MIDDLE  ALT=\"m=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the derivative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx=3x%5E2%2B2x%2B5 ALIGN=MIDDLE  ALT=\"dP%2Fdx=3x%5E2%2B2x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which has no real zeros.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx=3x%5E2%2B2x%2B5%3E0 ALIGN=MIDDLE  ALT=\"dP%2Fdx=3x%5E2%2B2x%2B5%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all 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;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E3%2Bx%5E2%2B5x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a continuously increasing function that must have exactly one real zero.<BR>\n" );
document.write( "That zero must be between <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=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-1%29=%28-1%29%5E3%2B%28-1%29%5E2%2B5%28-1%29%2B3=-1%2B1-5%2B3=-2 ALIGN=MIDDLE  ALT=\"P%28-1%29=%28-1%29%5E3%2B%28-1%29%5E2%2B5%28-1%29%2B3=-1%2B1-5%2B3=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=0%5E3%2B0%5E2%2B5%280%29%2B3=3 ALIGN=MIDDLE  ALT=\"P%280%29=0%5E3%2B0%5E2%2B5%280%29%2B3=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-2%2F3 ALIGN=MIDDLE  ALT=\"m=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%282%28-2%2F3%29-1%29x%5E2%2B%283%28-2%2F3%29%2B2%29x%2B3=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we had not found the three roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=0 ALIGN=MIDDLE  ALT=\"P%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so easily,<BR>\n" );
document.write( "we would look at the derivative for information,.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx=-2x%5E2%2B2%28-7%2F3%29x=-2x%28x%2B7%2F3%29 ALIGN=MIDDLE  ALT=\"dP%2Fdx=-2x%5E2%2B2%28-7%2F3%29x=-2x%28x%2B7%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which has the real zeros <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7%2F3 ALIGN=MIDDLE  ALT=\"x=-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-7%2F3 ALIGN=MIDDLE  ALT=\"x%3C-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx%3C0 ALIGN=MIDDLE  ALT=\"dP%2Fdx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > decreases.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%2F3%3Cx%3C0 ALIGN=MIDDLE  ALT=\"-7%2F3%3Cx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdx%3E0 ALIGN=MIDDLE  ALT=\"dP%2Fdx%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3 ALIGN=MIDDLE  ALT=\"P%28x%29=%28-2%2F3%29x%5E3%2B%28-7%2F3%29x%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > looks like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C4%2C8%2C3%2C9%2C-x%5E3%2B18x%5E2-107x%2B216%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C4%2C8%2C3%2C9%2C-x%5E3%2B18x%5E2-107x%2B216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There is a maximum at <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;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=3 ALIGN=MIDDLE  ALT=\"P%280%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7%2F3 ALIGN=MIDDLE  ALT=\"x=-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-7%2F3%29%3C0 ALIGN=MIDDLE  ALT=\"P%28-7%2F3%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , there are 3 zeros.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-7%2F3%29=%28-2%2F3%29%28-7%2F3%29%5E3%2B%28-7%2F3%29%28-7%2F3%29%5E2%2B3=%28-2%2F3%29%28-7%2F3%29%5E3%2B%28-7%2F3%29%5E3%2B3=%28-2%2F3%2B1%29%28-7%2F3%29%5E3%2B3=%281%2F3%29%28-343%2F27%29%2B3=-343%2F81%2B243%2F81=-100%2F81%3C0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so there are 3 zeros:<BR>\n" );
document.write( "one at some point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-7%2F3 ALIGN=MIDDLE  ALT=\"x%3C-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "another one for some point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%2F3%3Cx%3C0 ALIGN=MIDDLE  ALT=\"-7%2F3%3Cx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and one for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );