document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86297>Question 86297</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with this homework problem.  It says:  find all real zeros of the function<BR>\n" );
document.write( "a)  f(x)=-9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >+729<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b)  f(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >-11<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ 26x - 16\r<BR>\n" );
document.write( "\n" );
document.write( "The lecture notes are not very good on this and I can't understand the book.  The question is not from the book. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62391 by </B><A HREF=/tutors/aboutme.mpl?userid=scianci><B>scianci(186)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=scianci><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=62391\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)  f(x)=-9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >+729<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b)  f(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >-11<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ 26x - 16\r<BR>\n" );
document.write( "\n" );
document.write( "For a), begin by putting it in factored form:<BR>\n" );
document.write( "-9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >+729<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > - 81) = -9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(x + 9)(x - 9).<BR>\n" );
document.write( "Then, set each factor = 0. Each one will generate a solution:<BR>\n" );
document.write( "-9<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0 , x + 9 = 0 , x - 9 = 0<BR>\n" );
document.write( "x = 0 , x = -9 , x = 9. These are your real zeros.\r<BR>\n" );
document.write( "\n" );
document.write( "For b), apply the rational root theorem and look for a rational root first. By the rational root theorem, the numerator of any rational root will be a factor of 16 [which is your constant term] and the denominator of any rational root will be a factor of 1 [which is your lead coefficient]. So, any rational roots will be from among 1 , -1 , 2 , -2 , 4 , -4 , 8 , -8 , 16 , -16. Try each one in turn until you find one that fits the equation. Start with 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E3 ALIGN=MIDDLE  ALT=\"1%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >-11<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2 ALIGN=MIDDLE  ALT=\"1%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ 26(1) - 16 =<BR>\n" );
document.write( "1 - 11 + 26 - 16 = 0. So, x = 1 is a real zero and (x - 1) is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >-11<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ 26x - 16. Use synthetic diviison to locate its companion quadratic factor:<BR>\n" );
document.write( "1 1 -11  26 -16<BR>\n" );
document.write( "      1 -10  16<BR>\n" );
document.write( "  1 -10  16  0\r<BR>\n" );
document.write( "\n" );
document.write( "So,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >-11<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ 26x - 16 = (x - 1)((<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > - 10x + 16). Now, break down the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > - 10x + 16 into (x - 2)(x - 8). So, the factored form is:<BR>\n" );
document.write( "(x - 1)(x - 2)(x - 8).  Set each of these equal to 0 in turn:<BR>\n" );
document.write( "x - 1 = 0 , x - 2 = 0 , x - 8 = 0<BR>\n" );
document.write( "x = 1 , x = 2 , x = 8 are the real zeros. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );