document.write( "Question 570679: Find a 4th degree polynomial equation with integer coefficients which has
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Algebra.Com's Answer #367733 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Find a 4th degree polynomial equation with integer coefficients which has \r\n" );
document.write( "two irrational roots, one of which is 2+3\"sqrt%285%29\", and two imaginary\r\n" );
document.write( "roots, one of which is 3-2i.\r\n" );
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document.write( "In order to have integer coefficients if a polynomial equation has\r\n" );
document.write( "the irrational root \"A%2BB%2Asqrt%28C%29\", it must also have its conjugate\r\n" );
document.write( "\"A-B%2Asqrt%28C%29\".  Similarly if it has an imaginary root C+Di as a root,\r\n" );
document.write( "it must also have its conjugate C-Di as a solution. \r\n" );
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document.write( "So, since it must have integer coefficients, and 2+3\"sqrt%285%29\" is a root,\r\n" );
document.write( "then 2-3\"sqrt%285%29\" is also a root.  And since it has 3-2i as a root,\r\n" );
document.write( "3+2i is also a root.  So if it were solved we would have to end up with\r\n" );
document.write( "these four solutions:\r\n" );
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document.write( "x = 2+3\"sqrt%285%29\", x = 2-3\"sqrt%285%29\", x = 3-2i, x = 3+2i\r\n" );
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document.write( "So before that we would have had\r\n" );
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document.write( "x - (2+3\"sqrt%285%29\") = 0, x - (2-3\"sqrt%285%29\") = 0, x - (3-2i) = 0, x - (3+2i) = 0\r\n" );
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document.write( "And before that we would have had:\r\n" );
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document.write( "x - 2 - 3\"sqrt%285%29\" = 0, x - 2 + 3\"sqrt%285%29\" = 0, x - 3 + 2i = 0, x - 3 - 2i = 0\r\n" );
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document.write( "And before that we would have had:\r\n" );
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document.write( "(x - 2 - 3\"sqrt%285%29\")(x - 2 + 3\"sqrt%285%29\")(x - 3 + 2i)(x - 3 - 2i) = 0\r\n" );
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document.write( "Now we must multiply that out and collect terms to get our fourth\r\n" );
document.write( "degree polynomial equation:\r\n" );
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document.write( "It will be easier if we group the first two terms in each parentheses:\r\n" );
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document.write( "[(x-2) - 3\"sqrt%285%29\"][(x-2) + 3\"sqrt%285%29\"][(x-3) + 2i][(x-3) - 2i] = 0\r\n" );
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document.write( "The first two bracket expressions are like multiplying (A+B)(A-B)=A²-B², and\r\n" );
document.write( "the last two bracketed expressions are too:\r\n" );
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document.write( "[(x-2)² - 9·5][(x-3)² - 4i²] = 0\r\n" );
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document.write( "[(x-2)² - 45][(x-3)² - 4(-1)] = 0\r\n" );
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document.write( "[(x-2)² - 45][(x-3)² + 4] = 0\r\n" );
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document.write( "[x²-4x+4 - 45][x²-6x+9 + 4] = 0\r\n" );
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document.write( "(x² - 4x - 41)(x² - 6x + 13) = 0\r\n" );
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document.write( "x4 - 6x³ + 13x² - 4x³ + 24x² - 52x - 41x² + 246x - 533 = 0\r\n" );
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document.write( "x4 - 10x³ - 4x² + 194x - 533 = 0\r\n" );
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document.write( "Edwin

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