document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=253571>Question 253571</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial P(x) of degree 4 with zeros 2,2, 1-&#8730;3i, and with constant coefficient of 32.\r<BR>\n" );
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document.write( "Note: The third zero is one minus radical 3 times i. Which is an imaginary number known as radical negative one. \r<BR>\n" );
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document.write( "I am only given 3 zeros but I don't know the fourth zero! Can I assume the fourth zero is just x? \r<BR>\n" );
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document.write( "I tried doing the problem:<BR>\n" );
document.write( "P(x)= 32 (x-2) (x-2) [x- (1-&#8730;3i)]<BR>\n" );
document.write( "I got stuck from here. Please help me! Thank you for your time!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #185915 by </B><A HREF=/tutors/aboutme.mpl?userid=drk><B>drk(1908)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drk><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=185915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have 2 zeros at 2 and 2.<BR>\n" );
document.write( "The third zero is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+isqrt%283%29 ALIGN=MIDDLE  ALT=\"1+-+isqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The fourth zero is the conjugat of this which is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+isqrt%283%29 ALIGN=MIDDLE  ALT=\"1+%2B+isqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Remember: imaginaries always occur in pairs.<BR>\n" );
document.write( "So, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x-2%29%28x-2%29%28x+-+%281+-+isqrt%283%29%29%29%28x+-+%281+%2B+isqrt%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The product of these factors is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+%28x%5E2+-4x+%2B+4%29%28x%5E2+-2x+%2B+2%29 ALIGN=MIDDLE  ALT=\"P%28x%29+=+%28x%5E2+-4x+%2B+4%29%28x%5E2+-2x+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >and then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+X%5E4+-10x%5E3+%2B+14x%5E2+-16x+%2B+8 ALIGN=MIDDLE  ALT=\"P%28x%29+=+X%5E4+-10x%5E3+%2B+14x%5E2+-16x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But we wanted 32 as the constant, so multiply by 4 to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+4X%5E4+-40x%5E3+%2B+56x%5E2+-64x+%2B+32 ALIGN=MIDDLE  ALT=\"P%28x%29+=+4X%5E4+-40x%5E3+%2B+56x%5E2+-64x+%2B+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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