document.write( "Question 407894: Hi
\n" ); document.write( " Finding this difficult, new to this can someone please help\r
\n" ); document.write( "\n" ); document.write( "If z=2+j is one root of the equation z^4-2z^3-z^2+2z+10=0 find the other roots\r
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\n" ); document.write( "Karl \n" ); document.write( "
Algebra.Com's Answer #287422 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "z=2+j is one root of the equation z4 - 2z³- z²+ 2z + 10 = 0\r\n" );
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document.write( "We use synthetic division:\r\n" );
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document.write( "   2+j|1    -2      -1         2         10\r\n" );
document.write( "      |      2+j    -1+2j     -6+2j     -10                \r\n" );
document.write( "       1       j    -2+2j     -4+2j       0\r\n" );
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document.write( "So we have factored the polynomial as\r\n" );
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document.write( " [z - (2+j)][z³+ jz²+ (-2+2j)z + (-4+2j)z] = 0\r\n" );
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document.write( "Next we factor the cubic polynomial, since we know that if\r\n" );
document.write( "2+j is a root, its conjugate 2-j is also a root\r\n" );
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document.write( "   2-j|1       j    -2+2j     -4+2j\r\n" );
document.write( "      |      2-j     4-2j      4-2j\r\n" );
document.write( "       1     2       2         0\r\n" );
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document.write( "So we have now factored the polynomial as\r\n" );
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document.write( " [z - (2+j)][z - (2-j)](z²+ 2z + 2) = 0\r\n" );
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document.write( "We set each equal to 0\r\n" );
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document.write( "Setting the first two factors = 0 just gives z = 2+j and z = 2-j.\r\n" );
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document.write( "Setting the third factor = 0 gives\r\n" );
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document.write( "     z²+ 2z + 2 = 0\r\n" );
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document.write( "\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"\r\n" );
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document.write( "\"z+=+%28-2+%2B-+sqrt%28+2%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29+\"\r\n" );
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document.write( "\"z+=+%28-2+%2B-+sqrt%284-8%29%29%2F2+\"\r\n" );
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document.write( "\"z+=+%28-2+%2B-+sqrt%28-4%29%29%2F2+\"\r\n" );
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document.write( "\"z+=+%28-2+%2B-+2j%29%2F2+\"\r\n" );
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document.write( "\"z+=+2%28-1+%2B-+j%29%2F2+\"\r\n" );
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document.write( "\"z+=+cross%282%29%28-1+%2B-+j%29%2Fcross%282%29+\"\r\n" );
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document.write( "z = -1±j\r\n" );
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document.write( "So the roots other than the given one, 2+j, are 2-j, -1+j and -1-j\r\n" );
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document.write( "Edwin
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