document.write( "Question 60256: I need help with \"Find a polynomial of lowest degree with interger coefficients that has the indicated zeroes: 1+i,1-i,1,3\" \n" ); document.write( "
Algebra.Com's Answer #41367 by Edwin McCravy(20055)\"\" \"About 
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I need help with \"Find a polynomial of lowest degree\r\n" );
document.write( "with interger coefficients that has the indicated zeroes:\r\n" );
document.write( "1+i,1-i,1,3\"\r\n" );
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document.write( "Rule:\r\n" );
document.write( "For a polynomial to have a zero of r, it must have \r\n" );
document.write( "(x - r) as a factor.\r\n" );
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document.write( "Therefore,\r\n" );
document.write( "For the desired polynomial to have a zero of 1+i, it must have \r\n" );
document.write( "[x - (1+i)] as a factor.\r\n" );
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document.write( "For the desired polynomial to have a zero of 1-i, it must have \r\n" );
document.write( "[x - (1-i)] as a factor.\r\n" );
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document.write( "For the desired polynomial to have a zero of 1, it must have \r\n" );
document.write( "(x - 1) as a factor.\r\n" );
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document.write( "For the desired polynomial to have a zero of 3, it must have \r\n" );
document.write( "(x - 3) as a factor.\r\n" );
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document.write( "Therefore the desired polynomial must be equivalent to:\r\n" );
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document.write( "[x - (1+i)][x - (1-i)](x - 1)(x - 3)\r\n" );
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document.write( "Removing the inner parentheses in the first two factors:\r\n" );
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document.write( "[x - 1 - i][x - 1 + i](x - 1)(x - 3)\r\n" );
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document.write( "Grouping the (x - 1) in parentheses:\r\n" );
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document.write( "[(x - 1) - i][(x - 1) + i](x - 1)(x - 3)\r\n" );
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document.write( "Using FOIL on the first two factors:\r\n" );
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document.write( "[(x-1)2 + i(x - 1) - i(x - 1) - i2](x - 1)(x - 3)\r\n" );
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document.write( "The two middle terms in the brackets, i(x - 1) and -i(x - 1)\r\n" );
document.write( "cancel, and i2 = -1\r\n" );
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document.write( "[(x-1)2 - (-1)](x - 1)(x - 3)\r\n" );
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document.write( "[(x-1)(x-1) + 1)](x - 1)(x - 3)\r\n" );
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document.write( "[x2-2x+1 + 1](x - 1)(x - 3)\r\n" );
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document.write( "[x2 - 2x + 2](x - 1)(x - 3)\r\n" );
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document.write( "Use FOIL on the last two factors\r\n" );
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document.write( "[x2 - 2x + 2](x2 - 4x + 3)\r\n" );
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document.write( "Group the first two terms in each factor:\r\n" );
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document.write( "[(x2-2x) + 2][(x2-4x) + 3]\r\n" );
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document.write( "Use FOIL\r\n" );
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document.write( "(x2-2x)(x2-4x) + 3(x2-2x) + 2(x2-4x) - 6\r\n" );
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document.write( "Use FOIL on first term, remove parentheses\r\n" );
document.write( "in 2nd and 3rd terms\r\n" );
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document.write( "x4 - 6x3 + 8x3 + 3x2 - 6x + 2x2 - 8x + 6\r\n" );
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document.write( "x4 - 6x3 + 13x2 - 14x + 6  \r\n" );
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document.write( "Edwin

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