document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023290>Question 1023290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial function of least degree having only one real coefficients, a leading coefficient of 1, and roots 1-square root 3 , 1+square root 3, and 6-i .  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #638816 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=638816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The roots we know of are the real roots<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=1-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x%5B1%5D=1-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=1%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"x%5B2%5D=1%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the complex root <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B3%5D=6-i ALIGN=MIDDLE  ALT=\"x%5B3%5D=6-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For the polynomial to have only real coefficients,<BR>\n" );
document.write( "it must also have the conjugate complex root <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B4%5D=6%2Bi ALIGN=MIDDLE  ALT=\"x%5B4%5D=6%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the polynomial of least degree, with a leading coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%281%29 ALIGN=MIDDLE  ALT=\"red%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=red%281%29%2A%28x-x%5B1%5D%29%2A%28x-x%5B2%5D%29%2A%28x-x%5B3%5D%29%2A%28x-x%5B4%5D%29 ALIGN=MIDDLE   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-%281-sqrt%283%29%29%29%2A%28x-%281%2Bsqrt%283%29%29%29%2A%28x-%286-i%29%29%2A%28x-%286%2Bi%29%29 ALIGN=MIDDLE   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-1%2Bsqrt%283%29%29%2A%28x-1-sqrt%283%29%29%2A%28x-6%2Bi%29%2A%28x-6-i%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28x-1%2Bsqrt%283%29%29%2A%28x-1-sqrt%283%29%29%2A%28x-6%2Bi%29%2A%28x-6-i%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=%28%28x-1%29%2Bsqrt%283%29%29%2A%28%28x-1%29-sqrt%283%29%29%2A%28%28x-6%29%2Bi%29%2A%28%28x-6%29-i%29 ALIGN=MIDDLE   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=%28%28x-1%29%5E2-%28sqrt%283%29%29%5E2%29%2A%28%28x-6%29%5E2-i%5E2%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28%28x-1%29%5E2-%28sqrt%283%29%29%5E2%29%2A%28%28x-6%29%5E2-i%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=%28%28x%5E2-2x%2B1%29-3%29%2A%28%28x%5E2-12x%2B36%29-%28-1%29%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28%28x%5E2-2x%2B1%29-3%29%2A%28%28x%5E2-12x%2B36%29-%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%5E2-2x-2%29%2A%28%28x%5E2-12x%2B36%29%2B1%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28x%5E2-2x-2%29%2A%28%28x%5E2-12x%2B36%29%2B1%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%5E2-2x-2%29%2A%28x%5E2-12x%2B37%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28x%5E2-2x-2%29%2A%28x%5E2-12x%2B37%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From here on, it is just busywork.<BR>\n" );
document.write( "That is the part where I get bored and distracted, and make mistakes, so check my math from this point on.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E2%2A%28x%5E2-12x%2B37%29-2x%2A%28x%5E2-12x%2B37%29-2%2A%28x%5E2-12x%2B37%29 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E2%2A%28x%5E2-12x%2B37%29-2x%2A%28x%5E2-12x%2B37%29-2%2A%28x%5E2-12x%2B37%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%5E4-12x%5E3%2B37x%5E2%29-%282x%5E3-24x%5E2%2B74x%29-%282x%5E2-24x%2B74%29 ALIGN=MIDDLE  ALT=\"P%28x%29=%28x%5E4-12x%5E3%2B37x%5E2%29-%282x%5E3-24x%5E2%2B74x%29-%282x%5E2-24x%2B74%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=x%5E4-12x%5E3%2B37x%5E2-2x%5E3%2B24x%5E2-74x-2x%5E2%2B24x-74 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E4-12x%5E3%2B37x%5E2-2x%5E3%2B24x%5E2-74x-2x%5E2%2B24x-74\"   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=x%5E4-14x%5E3%2B59x%5E2-50x-74 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E4-14x%5E3%2B59x%5E2-50x-74\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );