document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1037022>Question 1037022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, can you state the possible number of imaginary zeros for these two functions?\r<BR>\n" );
document.write( "\n" );
document.write( "1. f(x) = ­-x³ ­-x² + 14x - 24\r<BR>\n" );
document.write( "\n" );
document.write( "2. f(x) = 2x³ ­- x² + 16x - 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651817 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=651817\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For polynomial functions with real coefficients, imaginary zeros come in pairs, so a polynomial function with real coefficients can have 0,2,4, 6,...imaginary zeros.<BR>\n" );
document.write( "However, a polynomial with degree 3 cannot have more than 3 zeros,<BR>\n" );
document.write( "so these functions can have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280%29 ALIGN=MIDDLE  ALT=\"highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282%29 ALIGN=MIDDLE  ALT=\"highlight%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > imaginary zeros.<BR>\n" );
document.write( "Polynomials of odd degree must have at least 1 real zero.<BR>\n" );
document.write( "They could have 1, 3, 5, all the way up to their degree.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "Each of those functions happens to have exactly 1 real zero.<BR>\n" );
document.write( "Some calculus plus calculations (or a graphing calculator) would tell you that.<BR>\n" );
document.write( "Since they are cubic (degree=3) polynomials, they must have a total of 3 zeros,<BR>\n" );
document.write( "so the other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-1=2 ALIGN=MIDDLE  ALT=\"3-1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > zeros are imaginary.<BR>\n" );
document.write( "You are probably not expected to be able to reach that conclusion, though. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );