document.write( "Question 1127861: Factor the polynomial function over the complex numbers.\r
\n" ); document.write( "\n" ); document.write( " $\"+f%28x%29+=++x%5E4+-+x%5E3+-+2x+-+4+\"$ \r
\n" ); document.write( "\n" ); document.write( "f(x) = \n" ); document.write( "

Algebra.Com's Answer #744304 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "Remember that \"complex numbers\" includes real numbers. So look for rational roots first.

\n" ); document.write( "Substitution show f(-1)=0, so -1 is a root. Extract that root using synthetic division.

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document.write( "   -1 |  1 -1  0 -2 -4\r\n" );
document.write( "      |    -1  2 -2  4\r\n" );
document.write( "      ----------------\r\n" );
document.write( "        1  -2  2 -4  0\r\n" );
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\n" ); document.write( "The remaining polynomial is $\"x%5E3-2x%5E2%2B2x-4\"$ . A second real root can be found using factoring by grouping.

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\n" ); document.write( "The remaining quadratic factor can be factored over the complex numbers as

\n" ); document.write( " $\"%28x%2Bi%2Asqrt%282%29%29%28x-i%2Asqrt%282%29%29\"$

\n" ); document.write( "So the complete factorization over the complex numbers is

\n" ); document.write( " $\"x%5E4-x%5E3-2x-4+=+%28x%2B1%29%28x-2%29%28x%2Bi%2Asqrt%282%29%29%28x-i%2Asqrt%282%29%29\"$
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