document.write( "Question 1065025: If z=1+5i is a solution to the equation: z^3 +az + b=0,
\n" ); document.write( "A) Find a and b if they are real numbers. For this part I got a=22 and b=-8
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Algebra.Com's Answer #680160 by Edwin McCravy(20081)\"\" \"About 
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document.write( "Looks like you got \"a\" correct but \"b\" incorrect.\r\n" );
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document.write( "We have to find B) and C) first in order to find A)!!\r\n" );
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document.write( "z³+az+b = 0\r\n" );
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document.write( "z³+0z²+az+b = 0\r\n" );
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document.write( "There are 3 roots to a 3rd degree equation:\r\n" );
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document.write( "Since the coefficient of z³ is 1, and the\r\n" );
document.write( "coefficient of z² is 0,\r\n" );
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document.write( "the sum of the roots is 0.  <-- answer to C)\r\n" );
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document.write( "Since 1+5i is a root, and since all coefficients are\r\n" );
document.write( "real, 1-5i is also a root.\r\n" );
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document.write( "Let the 3rd root be c\r\n" );
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document.write( "Sum of roots = (1+5i)+(1-5i)+c = 0\r\n" );
document.write( "                           2+c = 0\r\n" );
document.write( "                             c = -2\r\n" );
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document.write( "So the three roots are 1+5i, 1-5i, and -2  <-- answer to B).\r\n" );
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document.write( "We substitute -2 for z in the original:\r\n" );
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document.write( "      z³+az+b = 0\r\n" );
document.write( "(-2)³+a(-2)+b = 0\r\n" );
document.write( "      -8-2a+b = 0\r\n" );
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document.write( "Since the constant term b is the product of the roots\r\n" );
document.write( "with the opposite sign,\r\n" );
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document.write( "Product of roots = -(1+5i)(1-5i)(-2) = b\r\n" );
document.write( "                       -(1-25i²)(-2) = b\r\n" );
document.write( "                     -[1-25(-1)](-2) = b\r\n" );
document.write( "                         -[1+25](-2) = b\r\n" );
document.write( "                             -26(-2) = b\r\n" );
document.write( "                                  52 = b\r\n" );
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document.write( "We substitute b = 52 in -8a-2a+b = 0:\r\n" );
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document.write( "      -8-2a+b = 0\r\n" );
document.write( "     -8-2a+52 = 0\r\n" );
document.write( "       -2a+44 = 0\r\n" );
document.write( "          -2a = -44\r\n" );
document.write( "            a = 22 \r\n" );
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document.write( "We have already answered B) and C)\r\n" );
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document.write( "Edwin

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