document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=145754>Question 145754</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do I solve: y^4-3y^3-2y^2+10y-12=0, given root 1+i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #106368 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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=106368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I am going to switch notation from y to x.<BR>\n" );
document.write( "Since 1+i is a zero, 1-i will be as well.<BR>\n" );
document.write( "Thus we have:\r<BR>\n" );
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document.write( "(x-(1+i))(x-(1-i))(ax^2+bx+c)=0<BR>\n" );
document.write( "c - c i^2 + b x - 2 c x - b i^2 x + a x^2 - 2 b x^2 + c x^2 - <BR>\n" );
document.write( " a i^2 x^2 - 2 a x^3 + b x^3 + a x^4=0\r<BR>\n" );
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document.write( "Now, the coefficients of the original 4th degree polynomial are 1, -3, -2, 10,-12.\r<BR>\n" );
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document.write( "SO,<BR>\n" );
document.write( "2c=-12 implies c=-6.<BR>\n" );
document.write( "2b-2c=10 implies b=-1<BR>\n" );
document.write( "2a-2b+c=-2 implies a=1. \r<BR>\n" );
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document.write( "This gives (x-(1+i))(x-(1-i))(x^2-x-6)=0.<BR>\n" );
document.write( "We can solve for the other two zeroes by factoring.\r<BR>\n" );
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document.write( "(x-3)(x+2)=0<BR>\n" );
document.write( "gives x=3 or x=-2\r<BR>\n" );
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document.write( "So, x=3,x=-2,x=1 +i, x=1 -i  are the roots.\r<BR>\n" );
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document.write( "We can see this in the graph (only the real roots of 3 and -2 will show)...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E4-3x%5E3-2x%5E2%2B10x-12%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E4-3x%5E3-2x%5E2%2B10x-12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "As with my last answer, you can E-mail me if you would like the problem done in a different way. </SPAN>\n" );
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