document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107254>Question 107254</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need all the roots and zeros to these problems<BR>\n" );
document.write( "x^3+3x^2-5x-4=0<BR>\n" );
document.write( "x^4-2x^3-7x^2+18x-18+=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #78114 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=78114\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First example. <BR>\n" );
document.write( "A quick graph of the function shows that x=-4 looks like a root of the equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E3%2B3x%5E2-5x-4%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E3%2B3x%5E2-5x-4%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=x%5E3%2B3x%5E2-5x-4 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B3x%5E2-5x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=%28-4%29%5E3%2B3%2A%28-4%29%5E2-5%2A%28-4%29-4 ALIGN=MIDDLE  ALT=\"f%284%29=%28-4%29%5E3%2B3%2A%28-4%29%5E2-5%2A%28-4%29-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=-64%2B48%2B20-4 ALIGN=MIDDLE  ALT=\"f%284%29=-64%2B48%2B20-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=0 ALIGN=MIDDLE  ALT=\"f%284%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can then find the polynomial g(x) such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=g%28x%29%2A%28x%2B4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=g%28x%29%2A%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"g%28x%29=ax%5E2%2Bbx%2Bc\"   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=%28ax%5E2%2Bbx%2Bc%29%2A%28x%2B4%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28ax%5E2%2Bbx%2Bc%29%2A%28x%2B4%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=ax%5E3%2B%28b%2B4a%29x%5E2%2B%28c%2B4b%29%2Ax%2B4c ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E3%2B%28b%2B4a%29x%5E2%2B%28c%2B4b%29%2Ax%2B4c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Compare that to your original equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3%2B3x%5E2-5x-4 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3%2B3x%5E2-5x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you see that<BR>\n" );
document.write( "1.<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;\" ><BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B4a=3 ALIGN=MIDDLE  ALT=\"b%2B4a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2B4b=-5 ALIGN=MIDDLE  ALT=\"c%2B4b=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4c=-4 ALIGN=MIDDLE  ALT=\"4c=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From 1 and 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B4=3 ALIGN=MIDDLE  ALT=\"b%2B4=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From 4,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4c=-4 ALIGN=MIDDLE  ALT=\"4c=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1 ALIGN=MIDDLE  ALT=\"c=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x%5E2-x-1 ALIGN=MIDDLE  ALT=\"g%28x%29=x%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can then find the roots of g(x) using the quadratic equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   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-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The roots of g(x) are then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+sqrt%285%29%29%2F%282%29+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+sqrt%285%29%29%2F%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or approximately, x=-0.62 and x=1.62 \r<BR>\n" );
document.write( "\n" );
document.write( "Second example.<BR>\n" );
document.write( "A quick graph of the function shows that x=-3 and x=3 look like roots of the equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E4-2x%5E3-7x%5E2%2B18x-18%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+x%5E4-2x%5E3-7x%5E2%2B18x-18%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18 ALIGN=MIDDLE  ALT=\"h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-3%29=81%2B54-63-54-18 ALIGN=MIDDLE  ALT=\"h%28-3%29=81%2B54-63-54-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-3%29=0 ALIGN=MIDDLE  ALT=\"h%28-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18 ALIGN=MIDDLE  ALT=\"h%28x%29=x%5E4-2x%5E3-7x%5E2%2B18x-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%283%29=81-54-63%2B54-18 ALIGN=MIDDLE  ALT=\"h%283%29=81-54-63%2B54-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%283%29=0 ALIGN=MIDDLE  ALT=\"h%283%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then you can rewrite your equation to look like. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=g%28x%29%2A%28x%2B3%29%2A%28x-3%29 ALIGN=MIDDLE  ALT=\"h%28x%29=g%28x%29%2A%28x%2B3%29%2A%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"g%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Follow the first example above to find a,b,and c. <BR>\n" );
document.write( "Post another question if you get stuck. <BR>\n" );
document.write( "Good luck.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );