document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=42199>Question 42199</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534405991/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Factor the polynomial P(x)= 16x^4-81 completely and find all its zeros.  \r<BR>\n" );
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document.write( "I factored to (4x^2+9)(4x^2-9) then further to ((2x+3)(2x-3))^2 to get 3/2 and -3/2 both with multiplicity of 2 to equal four zeros.  But the solutions given also contain 3i/2 and -3i/2.  Also, I'm unsure that the second factoring step is appropriate since by distribution (2x+3)^2 is not equal to 4x^2+9 etc.  I'm stumped! Thanks for any help!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27320 by </B><A HREF=/tutors/aboutme.mpl?userid=psbhowmick><B>psbhowmick(878)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=psbhowmick><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=psbhowmick><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=27320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To avoid any error in this type of problems follow the following procedure.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=+16x%5E4-81 ALIGN=MIDDLE  ALT=\"P%28x%29=+16x%5E4-81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%29%5E2-9%5E2 ALIGN=MIDDLE  ALT=\"%284x%5E2%29%5E2-9%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B9%29%284x%5E2-9%29 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B9%29%284x%5E2-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B9%29%28%282x%29%5E2-3%5E2%29 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B9%29%28%282x%29%5E2-3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B9%29%282x%2B3%29%282x-3%29 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B9%29%282x%2B3%29%282x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This expression cannot be factorized as the formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2=%28a%2Bb%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=%28a%2Bb%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=%28a%2Bb%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Here you have commited error.\r<BR>\n" );
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document.write( "However, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B9 ALIGN=MIDDLE  ALT=\"4x%5E2%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factorized further using the concept of Complex Numbers.<BR>\n" );
document.write( "There you will find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i=sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e. 'i' is the imaginery square-root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2=-1 ALIGN=MIDDLE  ALT=\"i%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Multiplying both side by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9i%5E2=9 ALIGN=MIDDLE  ALT=\"-9i%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=-9i%5E2 ALIGN=MIDDLE  ALT=\"9=-9i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B9 ALIGN=MIDDLE  ALT=\"4x%5E2%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-9i%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2-9i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E2-%283i%29%5E2 ALIGN=MIDDLE  ALT=\"%282x%29%5E2-%283i%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3i%29%282x-3i%29 ALIGN=MIDDLE  ALT=\"%282x%2B3i%29%282x-3i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Thus, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%282x%2B3i%29%282x-3i%29%282x%2B3%29%282x-3%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%282x%2B3i%29%282x-3i%29%282x%2B3%29%282x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So the values of 'x' for which P(x) = 0 are: x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3i%2F2 ALIGN=MIDDLE  ALT=\"-3i%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3i%2F2 ALIGN=MIDDLE  ALT=\"3i%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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