document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=219501>Question 219501</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My math question is as follows:  Determine a polynomial P(x) with real coefficients and of degree 5 for which three of its roots are x=-1; x=1+2i; and x=-3i.  \r<BR>\n" );
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document.write( "I am setting my problem up as follows and wondering if this is correct:  P(x) = (x+1)[x+(1+2i)][x+(1-2i)][x+(3i)][x+(-3i)]  OR would the last 2 just be x+3i and x+(-3i).\r<BR>\n" );
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document.write( "Thanks Lori </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #165062 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=165062\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You are very close. Most of your +'s should be -'s. The general form would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+-+r%5B1%5D%29%28x+-+r%5B2%5D%29%28x+-+r%5B3%5D%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+-+r%5B1%5D%29%28x+-+r%5B2%5D%29%28x+-+r%5B3%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >... where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bn%5D ALIGN=MIDDLE  ALT=\"r%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a root. We want the minus signs so that when x is one of the roots, the minus makes the factor with that root after the minus sign zero which, in turn, makes the entire product zero.<br><BR>\n" );
document.write( "So your function should be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+-+%28-1%29%29%28x+-+%281+-+2i%29%29%28x+-+%281+%2B+2i%29%29%28x+-+%283i%29%29%28x+-+%28-3i%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying within the parentheses we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+%2B+1%29%28x+-+1+%2B+2i%29%28x+-+1+-+2i%29%28x+-+3i%29%28x+%2B+3i%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28x+-+1+%2B+2i%29%28x+-+1+-+2i%29%28x+-+3i%29%28x+%2B+3i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "To multiply this out take advantage of the pattern: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%28a+-+b%29+=+a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%28a+-+b%29+=+a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The second and third factors fit this pattern with \"a\" = \"x-1\" and \"b\" = 2i. The fourth and fifth factors also fit this pattern more obviously. So we'll start by multiplying the 2nd and 3rd factors and multiplying the 4th and 5th factors. (Remember multiplication is commutative and associative so we can multiply in any order we choose.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+%2B+1%29%28%28x+-+1%29%5E2+-+%282i%29%5E2%29%28x%5E2+-+%283i%29%5E2%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28%28x+-+1%29%5E2+-+%282i%29%5E2%29%28x%5E2+-+%283i%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+-+4i%5E2%29%28x%5E2+-+9i%5E2%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+-+4i%5E2%29%28x%5E2+-+9i%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=P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+-+4%28-1%29%29%28x%5E2+-+9%28-1%29%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+-+4%28-1%29%29%28x%5E2+-+9%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+%2B+4%29%28x%5E2+%2B+9%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+1+%2B+4%29%28x%5E2+%2B+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=P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+5%29%28x%5E2+%2B+9%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x+%2B+1%29%28x%5E2+-2x+%2B+5%29%28x%5E2+%2B+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point the i's have disappeared. We still have some multiplications to go. I think the easiest way is to multiply the 1st and third factors first:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x%5E3+%2B+x%5E2+%2B+9x+%2B+9%29%28x%5E2+-2x+%2B5%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x%5E3+%2B+x%5E2+%2B+9x+%2B+9%29%28x%5E2+-2x+%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And finally we multiply each term of the first factor by each term of the second factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+x%5E5+%2B+x%5E4+%2B+9x%5E3+%2B+9x%5E2+-+2x%5E4+-+2x%5E3+-+18x%5E2+-+18x+%2B+5x%5E3+%2B+5x%5E2+%2B+45x+%2B+45 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding like terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+x%5E5+-+x%5E4+%2B+12x%5E3+-+4x%5E2+%2B+27x+%2B+45 ALIGN=MIDDLE  ALT=\"P%28x%29+=+x%5E5+-+x%5E4+%2B+12x%5E3+-+4x%5E2+%2B+27x+%2B+45\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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