document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=670349>Question 670349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi I'm having trouble understanding. Finding all zero's of a polynomial function given one zero. My problem is: f(x)= x^4-32x^2-144; one zero is -2i <BR>\n" );
document.write( "The answer given is, 2i,6,-6 but I'm not understanding how. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #416893 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=416893\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the zeros of this polynomial function we need to set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factor the equation.\r<BR>\n" );
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document.write( "....since you are given  one zero which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%282i%29 ALIGN=MIDDLE  ALT=\"-%282i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is result of the square of negative number that could be only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-4%29 ALIGN=MIDDLE  ALT=\"sqrt%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-4%29 ALIGN=MIDDLE  ALT=\"sqrt%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you root <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2i ALIGN=MIDDLE  ALT=\"-2i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2i ALIGN=MIDDLE  ALT=\"2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > as well\r<BR>\n" );
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document.write( "therefore, you are actually given two roots, and now we will find remaining two\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+x%5E4-32x%5E2-144 ALIGN=MIDDLE  ALT=\"0=+x%5E4-32x%5E2-144\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32x%5E2 ALIGN=MIDDLE  ALT=\"-32x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-36x%5E2%2B4x%5E2 ALIGN=MIDDLE  ALT=\"-36x%5E2%2B4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-36%5E2%2B4x%5E2-144 ALIGN=MIDDLE  ALT=\"x%5E4-36%5E2%2B4x%5E2-144\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4-36%5E2%29%2B%284x%5E2-144%29 ALIGN=MIDDLE  ALT=\"%28x%5E4-36%5E2%29%2B%284x%5E2-144%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...factor out  <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;\" > from the first group and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%28x%5E2-36%29%2B4%28x%5E2-36%29 ALIGN=MIDDLE  ALT=\"x%5E2%28x%5E2-36%29%2B4%28x%5E2-36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B4%29%28x%5E2-36%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B4%29%28x%5E2-36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B4%29%28x-6%29%28x%2B6%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B4%29%28x-6%29%28x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we set each factor to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...............<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-4 ALIGN=MIDDLE  ALT=\"x%5E2=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .........=> given roots: the imaginary numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2i ALIGN=MIDDLE  ALT=\"-2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2i ALIGN=MIDDLE  ALT=\"2i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-6=0 ALIGN=MIDDLE  ALT=\"x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >..................real root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B6=0 ALIGN=MIDDLE  ALT=\"x%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............real root\r<BR>\n" );
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document.write( "show it on a graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C10%2C+-420%2C+200%2C%28x%5E2%2B4%29%28x-6%29%28x%2B6%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C10%2C+-420%2C+200%2C%28x%5E2%2B4%29%28x-6%29%28x%2B6%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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