document.write( "Question 1144093: Use synthetic division to find the quotient and remainder when dividing
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Algebra.Com's Answer #765089 by Edwin McCravy(20065) $\"\"$ $\"About$

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Use synthetic division to find the quotient and remainder when dividing
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document.write( "When there are imaginary numbers, the synthetic division is essentially the same\r\n" );
document.write( "process as when there are only real numbers.  There are basically just two\r\n" );
document.write( "differences:\r\n" );
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document.write( "1. We just need to skip some extra space after the real part of each coefficient\r\n" );
document.write( "to have room for the imaginary parts of the coefficients.  Put the imaginary\r\n" );
document.write( "parts out to the right of the real numbers: \r\n" );
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document.write( "2. Also we will have to do some scratch work to multiply complex numbers:\r\n" );
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document.write( "We start the same way as when there are only real numbers:\r\n" );
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document.write( "i | 1    1    -1    2   \r\n" );
document.write( "  |       +i            ` \r\n" );
document.write( "    1    \r\n" );
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document.write( "Then we add the 1 and the +i and write the sum 1+i below the line:\r\n" );
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document.write( "i | 1    1    -1    2   \r\n" );
document.write( "  |       +i            `   \r\n" );
document.write( "    1    1+i\r\n" );
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document.write( "We have to do some scratch work here to multiply 1+i by i:\r\n" );
document.write( "i(1+i) = i+i² = i+(-1) = -1+i\r\n" );
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document.write( "i | 1    1    -1    2   \r\n" );
document.write( "  |       +i  -1+i      `\r\n" );
document.write( "    1    1+i  -2+i\r\n" );
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document.write( "We do some more scratch work to multiply -2+i by i:\r\n" );
document.write( "i(-2+i) = -2i+i² = -2i+(-1) = -1-2i\r\n" );
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document.write( "i | 1    1    -1     2   \r\n" );
document.write( "  |       +i  -1+i  -1-2i`   \r\n" );
document.write( "    1    1+i  -2+i   1-2i\r\n" );
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document.write( "So the quotient is a polynomial which has degree which is\r\n" );
document.write( "1 less than the degree of the original polynomial.  The\r\n" );
document.write( "original polynomial has degree 3, so the quotient will have\r\n" );
document.write( "degree which is 1 less, so it will have degree 2.  Its\r\n" );
document.write( "coefficients are on the bottom line of the synthetic \r\n" );
document.write( "division all but the last complex number, which is the \r\n" );
document.write( "remainder.\r\n" );
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document.write( "So the answers are:\r\n" );
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document.write( "Quotient = 1x² + (1+i)x + (-2+i)  [or x² + x + ix - 2 + i]\r\n" );
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document.write( "Remainder = 1-2i\r\n" );
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document.write( "Edwin

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