document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186857>Question 1186857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use synthetic division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder. \r<BR>\n" );
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document.write( "2x^3 − 8ix^2 + 4x + (9−2i)/x−2i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849367 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to perform synthetic division with complex numbers and rewrite the expression:\r<BR>\n" );
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document.write( "**1. Set up the synthetic division:**\r<BR>\n" );
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document.write( "Write down the coefficients of the polynomial in the numerator and the value of x that makes the denominator zero (2i in this case):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |________________________<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**2. Perform the synthetic division:**\r<BR>\n" );
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document.write( "*   Bring down the first coefficient (2):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Multiply the value outside (2i) by the number just brought down (2) and write the result under the next coefficient (-8i):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Add the numbers in the second column (-8i and 4i):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Multiply 2i by -4i and write the result under the next coefficient (4):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i   8<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i   12<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Add the numbers in the third column (4 and 8):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i   8<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i   12<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Multiply 2i by 12 and write the result under the last term (9-2i):\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i   8   24i<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i   12   9 + 22i<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "*   Add the numbers in the last column:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "2i |  2   -8i    4    9 - 2i<BR>\n" );
document.write( "     |      4i   8   24i<BR>\n" );
document.write( "     ------------------------<BR>\n" );
document.write( "       2   -4i   12   9 + 22i<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**3. Interpret the results:**\r<BR>\n" );
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document.write( "The numbers in the bottom row (2, -4i, 12) are the coefficients of the quotient, and the last number (9 + 22i) is the remainder. The degree of the quotient is one less than the degree of the original polynomial.\r<BR>\n" );
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document.write( "Therefore, the quotient q(x) is 2x² - 4ix + 12, and the remainder r(x) is 9 + 22i. The divisor d(x) is x - 2i.\r<BR>\n" );
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document.write( "**4. Write the expression:**\r<BR>\n" );
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document.write( "The expression can be written as:\r<BR>\n" );
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document.write( "2x² - 4ix + 12 + (9 + 22i) / (x - 2i)<BR>\n" );
document.write( " </SPAN>\n" );
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