document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=533623>Question 533623</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please state the positive real zeros, negative real zeros, and imaginary zero for this functions : r(x)=x^4-x^3-5x^2+6x+1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352236 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=352236\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Can you please state the positive real zeros, negative real zeros, and imaginary zero for this functions : r(x)=x^4-x^3-5x^2+6x+1<BR>\n" );
document.write( "**<BR>\n" );
document.write( "Using the Rational Roots Theorem to find the zeros:\r<BR>\n" );
document.write( "\n" );
document.write( "....0....|....1........-1........-5.........6.........1<BR>\n" );
document.write( "....1....|....1..........0........-5.........1.........2<BR>\n" );
document.write( "....2....|....1......... 1........-3.........0.........1<BR>\n" );
document.write( "....3....|....1........  2........ 1.........6.........19 (3 is upper bound)\r<BR>\n" );
document.write( "\n" );
document.write( "....0....|....1........-1........-5.........6.........1<BR>\n" );
document.write( "..-1....|....1........-2........-3.........3........-2 (real root between 0 and -1)<BR>\n" );
document.write( "..-2....|....1........-3........1.........4.........-7<BR>\n" );
document.write( "..-3....|....1........-4........7......-15.........46 (-3 is lower bound)  (real root between-2 and -3)\r<BR>\n" );
document.write( "\n" );
document.write( "Procedure above shows there are two real roots between 0 and -1 and between -2 and -3,<BR>\n" );
document.write( "but they are not rational. I don't know how to find these two irrational real roots algebraically. The graphics program on my computer show these roots to be -0.2929 and -2.2740. The other two roots are non-real or imaginary. See graph below as a visual check.<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+-5%2C+5%2Cx%5E4-x%5E3-5x%5E2%2B6x%2B1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E4-x%5E3-5x%5E2%2B6x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );