document.write( "Question 852534: Given that 4-i is the root and {x-(4-i)}is a factor, use algebra (not your calculator) to solve the following equation: x^4-8x^3+19x^2-16x+34=0 \n" ); document.write( "

Algebra.Com's Answer #513488 by Edwin McCravy(20060) $\"\"$ $\"About$

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Given that 4-i is the root and {x-(4-i)}is a factor, use algebra (not your calculator) to solve the following equation: x^4-8x^3+19x^2-16x+34=0
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document.write( "Let's use synthetic division.  [Yes, you can use synthetic division\r\n" );
document.write( "even with complex numbers!  However we'll have to stop along the way\r\n" );
document.write( "to multiply two complex numbers.]\r\n" );
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document.write( "4-i | 1   -8   19   -16   34\r\n" );
document.write( "    |      4                 \r\n" );
document.write( "      1   -4-i\r\n" );
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document.write( "We stop and multiply -4-i by 4-i: (-4-i)(4-i) = -16+4i-4i+i² = -16+(-1) = -17\r\n" );
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document.write( "4-i | 1   -8    19  -16    34\r\n" );
document.write( "    |      4-1 -17    8-2i   \r\n" );
document.write( "      1   -4-i   2   -8-2i\r\n" );
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document.write( "We stop and multiply -8-2i by 4-i: (-8-2i)(4-i) = -32+8i-8i+2i² = -32+2(-1) = -34\r\n" );
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document.write( "4-i | 1   -8    19  -16     34\r\n" );
document.write( "    |      4-1 -17    8-2i -34\r\n" );
document.write( "      1   -4-i   2   -8-2i   0\r\n" );
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document.write( "We have factored the equation as\r\n" );
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document.write( "[x-(4-i)][x³ + (-4-i)x² + 2x +(-8-2i)] = 0\r\n" );
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document.write( "Now since 4-i is a root, so is its conjugate 4+i. Now we divide\r\n" );
document.write( "the bracketed third degree polynomial by [x-(4+i)]\r\n" );
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document.write( "4+i | 1   -4-i   2    -8-2i\r\n" );
document.write( "    |      4+i   0     8+2i\r\n" );
document.write( "      1      0   2        0\r\n" );
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document.write( "Now we have further factored the polynomial equation as\r\n" );
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document.write( "[x-(4-i)][x-(4+i)](x²+2)\r\n" );
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document.write( "We can factor x²+2 by writing it as x²-(-2) = x²-(√-2)² = (x-√-2)(x+√-2) = \r\n" );
document.write( "(x-i√2)(x+i√2)}}}\r\n" );
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document.write( "So the final factorization is:\r\n" );
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document.write( "[x-(4-i)][x-(4+i)](x-i√2)(x+i√2) = 0 \r\n" );
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document.write( "So the 4 solutions are 4-i, 4+i, i√2, -i√2\r\n" );
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document.write( "Edwin

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