document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194674>Question 194674</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131019635/algebrahomeworkh>An Introduction to Abstract Algebra with Notes to the Future Teacher</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> with the help of the quadratic formula, find all the complex roots of each of the polynomials given in exercise 1.\r<BR>\n" );
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document.write( "iii)   z^3 + i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #146065 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=146065\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, perform synthetic division on the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2Bi ALIGN=MIDDLE  ALT=\"z%5E3%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the test zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=i ALIGN=MIDDLE  ALT=\"z=i\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note: you can easily solve for \"z\" and find that one solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=i ALIGN=MIDDLE  ALT=\"z=i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "i |  1  0  0   i\r\n" );
document.write( "  |     i  -1  -i\r\n" );
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document.write( "     1  i  -1   0\r\n" );
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document.write( "Since the first three numbers in the third row are 1, i, and -1, this means that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2Bi=%28z-i%29%28z%5E2%2Biz-1%29 ALIGN=MIDDLE  ALT=\"z%5E3%2Bi=%28z-i%29%28z%5E2%2Biz-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now simply use the quadratic formula to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2%2Biz-1=0 ALIGN=MIDDLE  ALT=\"z%5E2%2Biz-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for \"z\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%28i%29+%2B-+sqrt%28+%28i%29%5E2-4%281%29%28-1%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%28i%29+%2B-+sqrt%28+%28i%29%5E2-4%281%29%28-1%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=i ALIGN=MIDDLE  ALT=\"b=i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1 ALIGN=MIDDLE  ALT=\"c=-1\"   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=z+=+%28-i+%2B-+sqrt%28+-1-4%281%29%28-1%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+%2B-+sqrt%28+-1-4%281%29%28-1%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2=%28sqrt%28-1%29%29%5E2=-1 ALIGN=MIDDLE  ALT=\"i%5E2=%28sqrt%28-1%29%29%5E2=-1\"   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=z+=+%28-i+%2B-+sqrt%28+-1%2B4+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+%2B-+sqrt%28+-1%2B4+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-i+%2B-+sqrt%28+3+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+%2B-+sqrt%28+3+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the other two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-i+%2B+sqrt%28+3+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+%2B+sqrt%28+3+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-i+-+sqrt%28+3+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+-+sqrt%28+3+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the 3 solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=i ALIGN=MIDDLE  ALT=\"z=i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-i+%2B+sqrt%28+3+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+%2B+sqrt%28+3+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-i+-+sqrt%28+3+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-i+-+sqrt%28+3+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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