document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=238136>Question 238136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I NEED HELP ASAP\r<BR>\n" );
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document.write( "Can you break this problem down for me so, I can do other problems like it\r<BR>\n" );
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document.write( "Find a polynomial function of lowest degree with rational coefficients that has the given numbers of some of its zeros.\r<BR>\n" );
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document.write( "2+i, 3\r<BR>\n" );
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document.write( "I know there is a conjugate of 2+i, which is 2-i. Which makes the degree of the polynomial 3 and the number of zeros 3\r<BR>\n" );
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document.write( "2+1,2-i,3\r<BR>\n" );
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document.write( "(x-(2+i))(x-(2-i))(x-3)\r<BR>\n" );
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document.write( "I can't get passed the imaginary numbers' portion of the problem. And if I do, it is wrong.\r<BR>\n" );
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document.write( "PLEASE HELP\r<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 #175017 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=175017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> two roots of this equation are:\r<BR>\n" );
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document.write( "x = 3<BR>\n" );
document.write( "x = 2+i\r<BR>\n" );
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document.write( "x - 3 = 0 \r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "x - 2 - i = 0\r<BR>\n" );
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document.write( "it seems you're on the right track with the conjugate of x = 2-i\r<BR>\n" );
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document.write( "that should make them cancel out when you multiply them together.\r<BR>\n" );
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document.write( "let's do that and see how it works.\r<BR>\n" );
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document.write( "x = 2-i means that x - 2 + i = 0\r<BR>\n" );
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document.write( "you have 3 factors.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "x - 3 = 0<BR>\n" );
document.write( "x - 2 + i = 0<BR>\n" );
document.write( "x - 2 - i = 0\r<BR>\n" );
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document.write( "first you want to multiply x - 2 + i by x - 2 - i\r<BR>\n" );
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document.write( "each term in one factor has to be multiplied by each term in the other factor.\r<BR>\n" );
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document.write( "start with x in the left factor and multiply each term in the right factor from left to right.\r<BR>\n" );
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document.write( "x * x = x^2<BR>\n" );
document.write( "x * -2 = -2x<BR>\n" );
document.write( "x * -i = -ix\r<BR>\n" );
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document.write( "do -2 next.\r<BR>\n" );
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document.write( "-2 * x = -2x<BR>\n" );
document.write( "-2 * -2 = 4<BR>\n" );
document.write( "-2 * -i = 2i\r<BR>\n" );
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document.write( "do i next.\r<BR>\n" );
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document.write( "i * x = ix<BR>\n" );
document.write( "i * -2 = -2i<BR>\n" );
document.write( "i * -i = -i^2 = 1 because i^2 = -1\r<BR>\n" );
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document.write( "add all your terms together to get:\r<BR>\n" );
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document.write( "x^2 - 2x - ix -2x + 4 + 2i + ix -2i + 1\r<BR>\n" );
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document.write( "+ ix and - ix cancel out<BR>\n" );
document.write( "+ 2i and - 2i cancel out<BR>\n" );
document.write( "-2x and -2x become -4x<BR>\n" );
document.write( "+ 4 and + 1 become + 5\r<BR>\n" );
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document.write( "your equation becomes x^2 - 4x + 5\r<BR>\n" );
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document.write( "multiply this equation by x-3 to get:\r<BR>\n" );
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document.write( "x * x^2 = x^3<BR>\n" );
document.write( "x * -4x = -4x^2<BR>\n" );
document.write( "x * 5 = 5x<BR>\n" );
document.write( "-3 * x^2 = -3x^2<BR>\n" );
document.write( "-3 * -4x = 12x<BR>\n" );
document.write( "-3 * 5 = -15\r<BR>\n" );
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document.write( "add these terms together to get:\r<BR>\n" );
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document.write( "x^3 - 4x^2 + 5x - 3x^2 + 12x - 15\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "x^3 -7x^2 + 17x - 15\r<BR>\n" );
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document.write( "that's your equation.\r<BR>\n" );
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document.write( "a graph of this equation looks like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-20%2C20%2C-20%2C20%2Cx%5E3+-+7x%5E2+%2B+17x+-+15%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-20%2C20%2C-20%2C20%2Cx%5E3+-+7x%5E2+%2B+17x+-+15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It only has one real root (x-axis crossing point).\r<BR>\n" );
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document.write( "If you set x = 3, the equation should become equal to 0.\r<BR>\n" );
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document.write( "I confirmed that it does.\r<BR>\n" );
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document.write( "If you set x = 2-i, the equation should become equal to 0.\r<BR>\n" );
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document.write( "I confirmed that it does.  It was a nosebleed but it was confirmed.\r<BR>\n" );
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document.write( "I suspect 2+i will confirm as well although I'm not going to go through that again.\r<BR>\n" );
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document.write( "Not today, anyway.\r<BR>\n" );
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document.write( "   \r<BR>\n" );
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document.write( " </SPAN>\n" );
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