document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1121240>Question 1121240</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to find a polynomial function with rational coefficients so that p(x)=0 has the given roots 2+i and 1-sqrt5\r<BR>\n" );
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document.write( "I tried (x-(2+i))(x-(2-i))(x-(1-sqrt5))(x-(1+sqrt5)) and got x^4-6x^3+9x^2+6x-20, which does not seem to include 2+i as a root. Am I doing it right?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #737039 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=737039\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> At the end, I got what you got.<BR>\n" );
document.write( "If one raises 2+i to the 4th power, the answer is -7+24 i<BR>\n" );
document.write( "to the third power is 2+ 11i<BR>\n" );
document.write( "squared is 3+4i<BR>\n" );
document.write( "The constants, after using the exponents, are -7-12+27+12-20<BR>\n" );
document.write( "The i s will be 24-66+36+6.  You did it right<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-50%2C50%2Cx%5E4-6x%5E3%2B9x%5E2%2B6x-20%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-50%2C50%2Cx%5E4-6x%5E3%2B9x%5E2%2B6x-20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Your answer\r<BR>\n" );
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document.write( "another way, longer, uses the quadratic formula to find where 2+i and 2-i work<BR>\n" );
document.write( "a=1<BR>\n" );
document.write( "b=-4, because will divide by 2<BR>\n" );
document.write( "sqrt(b^2-4ac)=2i, and when that is divided by 2, will give 1i as a result.  sqrt(16-20) does that, and since 4ac=20, c must equal 5<BR>\n" );
document.write( "One polynomial is x^2-4x+5\r<BR>\n" );
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document.write( "The other polynomial has <BR>\n" );
document.write( "a=1<BR>\n" );
document.write( "b=-2, so when divided by 2 will get -1<BR>\n" );
document.write( "sqrt(b^2-4ac)  needs to be 2sqrt(5), because we are dividing the numerator by 2<BR>\n" );
document.write( "b^2-4ac=20<BR>\n" );
document.write( "4-4c=20<BR>\n" );
document.write( "-4c=16<BR>\n" );
document.write( "c=-4<BR>\n" );
document.write( "x^2-2x-4 is the other polynomial\r<BR>\n" );
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document.write( "product is x^4-2x^3-4x^2<BR>\n" );
document.write( "==========-4x^3+8x^2+16x<BR>\n" );
document.write( "===============5x^2-10x-20<BR>\n" );
document.write( "or x^4-6x^3+9x^2+6x-20 </SPAN>\n" );
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