document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1003892>Question 1003892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial with real corfficients that has -2 and 2 + i as zeros </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #620508 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=620508\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = 2+i is a root, so x = 2-i is another root. Complex roots come in conjugate pairs\r<BR>\n" );
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document.write( "Let's focus on x = 2+i. Get everything to one side (so 0 is on the other side).\r<BR>\n" );
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document.write( "x = 2+i<BR>\n" );
document.write( "x - 2 = i<BR>\n" );
document.write( "(x-2)^2 = i^2<BR>\n" );
document.write( "(x-2)^2 = -1<BR>\n" );
document.write( "x^2-4x+4 = -1<BR>\n" );
document.write( "x^2-4x+4+1 = -1+1<BR>\n" );
document.write( "x^2-4x+5 = 0\r<BR>\n" );
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document.write( "So we see that if 2+i is a root of p(x), then (x^2-4x+5) is a factor of p(x).<BR>\n" );
document.write( "The same can be said about 2-i. <BR>\n" );
document.write( "The same basic steps are followed so I won't show them. \r<BR>\n" );
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document.write( "If x = -2 is a root of p(x), then x-(-2) = x+2 is a factor of p(x)\r<BR>\n" );
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document.write( "So we know that (x+2) and (x^2-4x+5) are factors of p(x)\r<BR>\n" );
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document.write( "Let's multiply the two factors\r<BR>\n" );
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document.write( "(x+2)(x^2-4x+5) = x(x^2-4x+5)+2(x^2-4x+5)<BR>\n" );
document.write( "(x+2)(x^2-4x+5) = x*x^2+x*(-4x)+x*5+2*x^2+2*(-4x)+2*5<BR>\n" );
document.write( "(x+2)(x^2-4x+5) = x^3-4x^2+5x+2x^2-8x+10<BR>\n" );
document.write( "(x+2)(x^2-4x+5) = x^3-2x^2-3x+10\r<BR>\n" );
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document.write( "So in the end, the polynomial is <font size=4 color=red>x^3-2x^2-3x+10</font> which is the final answer </SPAN>\n" );
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