document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166375>Question 166375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a rational zero of the polynomial function and use it to find all the zeros of the function.\r<BR>\n" );
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document.write( "f(x) = 2x^3 - x^2 - 12x + 6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122640 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=122640\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's graph the function.<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%2C+2x%5E3+-+x%5E2+-+12x+%2B+6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+2x%5E3+-+x%5E2+-+12x+%2B+6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looks like x=0.5 is a zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+x%5E2+-+12x+%2B+6=2%280.5%29%5E3-%280.5%29%5E2-12%280.5%29%2B6 ALIGN=MIDDLE  ALT=\"2x%5E3+-+x%5E2+-+12x+%2B+6=2%280.5%29%5E3-%280.5%29%5E2-12%280.5%29%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+x%5E2+-+12x+%2B+6=%280.5%29%5E2-%280.5%29%5E2-6%2B6 ALIGN=MIDDLE  ALT=\"2x%5E3+-+x%5E2+-+12x+%2B+6=%280.5%29%5E2-%280.5%29%5E2-6%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+x%5E2+-+12x+%2B+6=0 ALIGN=MIDDLE  ALT=\"2x%5E3+-+x%5E2+-+12x+%2B+6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Even though x=2.5 and x=-2.5 look like zeros, they don't solve.<BR>\n" );
document.write( "Let's use synthetic divsion to factor out (x-0.5) and get the quadratic equation that's left over.<BR>\n" );
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document.write( "Showing synthetic division is a little difficult.<BR>\n" );
document.write( "The first part will show the factor (left hand column), then the factor times the divisor,<BR>\n" );
document.write( "then the next line will show the subtraction.<BR>\n" );
document.write( "Repeat.<BR>\n" );
document.write( "Hopefully it makes sense.<BR>\n" );
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document.write( "...........___________________<BR>\n" );
document.write( "(x-0.5)|2x^3-x^2-12x+6 <BR>\n" );
document.write( "2x^2:....2x^3-x^2<BR>\n" );
document.write( ".................................-12x+6<BR>\n" );
document.write( "-12:..........................-12x+6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3-x%5E2-12x%2B6%29%2F%28x-0.5%29=2x%5E2-12 ALIGN=MIDDLE  ALT=\"%282x%5E3-x%5E2-12x%2B6%29%2F%28x-0.5%29=2x%5E2-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3-x%5E2-12x%2B6%29=%28x-0.5%29%2A%282x%5E2-12%29 ALIGN=MIDDLE  ALT=\"%282x%5E3-x%5E2-12x%2B6%29=%28x-0.5%29%2A%282x%5E2-12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-12=0 ALIGN=MIDDLE  ALT=\"2x%5E2-12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=6 ALIGN=MIDDLE  ALT=\"x%5E2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0+%2B-+sqrt%286%29 ALIGN=MIDDLE  ALT=\"x=0+%2B-+sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The three zeros are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%286%29 ALIGN=MIDDLE  ALT=\"-sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );