document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=993883>Question 993883</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you sketch the graph of a polynomial function that has 3 real zeros, and two imaginary zeros. Like how would you graph this polynomial function 3x^5-4x^4+2x^3-x^2-x-7. It has 3 real zeros and 2 imaginary,but I don't know how to graph that.<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 #613061 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=613061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you have to find the roots.<BR>\n" );
document.write( "the graph will cross or touch the x-axis at the real roots.<BR>\n" );
document.write( "the graph will not cross or touch the x-axis at the imaginary roots.<BR>\n" );
document.write( "if the root has an even multiplicity, the graph will touch the x-axis but not cross it.<BR>\n" );
document.write( "if the root has an odd multiplicity,the graph will cross the x-axis.<BR>\n" );
document.write( "in between the real roots, you just have to test selected points in order to be able to sketch the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "check this reference out as it explains how to sketch a graph and it also discusses even roots and odd roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of your equation looks like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C3x%5E5-4x%5E4%2B2x%5E3-x%5E2-x-7%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C3x%5E5-4x%5E4%2B2x%5E3-x%5E2-x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since it only crosses the x-axis once, it has only 1 real root.<BR>\n" );
document.write( "the other 4 must be complex.<BR>\n" );
document.write( "fyi - complex roots always come in pairs.<BR>\n" );
document.write( "you can have 2 or 4, but you can't have 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's a good reference that should help you understand better.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/\" target = \"_blank\">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "lots of good stuff in there.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "check out tutorials 38 and 39.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "they apply to your question.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "basically you are looking for the real roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "there are all kinds of tests to determine what they are.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "once you found those, if the number of roots is not equal to the degree of the equation, then the other roots must be complex.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );