document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=554217>Question 554217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you list the possible rational zeros of f in a function using the rational zero theorem?<BR>\n" );
document.write( "f(x)=x^4+2x^2-24 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #361289 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=361289\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> POSSIBLE RATIONAL ZEROS<BR>\n" );
document.write( "The rational zeros will be rational numbers (fractions) with denominators that are divisors/factors of the leading coefficient, and numerators that are divisors/factors of the independent term.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E4%2B2x%5E2-24 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E4%2B2x%5E2-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > the leading coefficient is the invisible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > in front of <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;\" >, and the independent term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24 ALIGN=MIDDLE  ALT=\"-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The only positive divisors/factor of 1 is 1. <BR>\n" );
document.write( "The positive divisors/factors of 24 are:<BR>\n" );
document.write( "1, 2, 3, 4, 6, 8, 12, and 24.<BR>\n" );
document.write( "The possible rational zeros are 1, -1, 2, -2, 3, -3, etc.<BR>\n" );
document.write( "HOW TO FIND THE POSSIBLE RATIONAL ZEROS<BR>\n" );
document.write( "My method to find the factors, was to start with 1, and check integers to see if they would divide 24 evenly. Once I got to a factor that squared was equal or greater than 24, I used another strategy.<BR>\n" );
document.write( "Factors come in pairs that multiply to give you 24:<BR>\n" );
document.write( "1 x 24 = 24,<BR>\n" );
document.write( "2 x 12 = 24<BR>\n" );
document.write( "3 x 8 = 24<BR>\n" );
document.write( "4 x 6 = 24<BR>\n" );
document.write( "Once I got to 6, and saw that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E2=36%3E24 ALIGN=MIDDLE  ALT=\"6%5E2=36%3E24\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "I found the larger factors from the smaller factors I found before, by dividing as in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F3=8 ALIGN=MIDDLE  ALT=\"24%2F3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F2=12 ALIGN=MIDDLE  ALT=\"24%2F2=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Another method to find the factors is to work form the prime factorization:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=2%5E3%2A3 ALIGN=MIDDLE  ALT=\"24=2%5E3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the factors/divisors will all be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ea%2A3%5Eb ALIGN=MIDDLE  ALT=\"2%5Ea%2A3%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=a%3C=3 ALIGN=MIDDLE  ALT=\"0%3C=a%3C=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=b%3C=1 ALIGN=MIDDLE  ALT=\"0%3C=b%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "There are 4 possible values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 2 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so I knew there would be 8 factors.<BR>\n" );
document.write( "For a number like 24, calculating them as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ea%2A3%5Eb ALIGN=MIDDLE  ALT=\"2%5Ea%2A3%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" > would have been a pain, so I used the other method.<BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "To find the zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E4%2B2x%5E2-24 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E4%2B2x%5E2-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I would solve the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B2x%5E2-24=0 ALIGN=MIDDLE  ALT=\"x%5E4%2B2x%5E2-24=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by changing variables to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so that the equation would transform into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B2y-24=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B2y-24=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-4%29%28y%2B6%29=0 ALIGN=MIDDLE  ALT=\"%28y-4%29%28y%2B6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Back to the original function, I would write it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E4%2B2x%5E2-24=%28x%5E2-4%29%28x%5E2%2B6%29=%28x-2%29%28x%2B2%29%28x%5E2%2B6%29 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E4%2B2x%5E2-24=%28x%5E2-4%29%28x%5E2%2B6%29=%28x-2%29%28x%2B2%29%28x%5E2%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then I would know that the only real zeros are 2, and -2, which were 2 of the 24 possible rational zeros.<BR>\n" );
document.write( "The factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B6%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not zero for any real value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );