document.write( "Question 102465: the degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots. Express f(x) as the productof of linear and quadriatic polynomials with real coefficients \n" ); document.write( "
Algebra.Com's Answer #74466 by Edwin McCravy(20059)\"\" \"About 
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the degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots. Express f(x) as the productof of linear and quadriatic polynomials with real coefficients\r
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document.write( "If a polynomial has all real coefficients and has A+Bi as a root,\r\n" );
document.write( "then its conjugate A-Bi is also a root of the the polynomial.\r\n" );
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document.write( "So since the polynomial we want has +4i as a root, then it also\r\n" );
document.write( "has -4i as a root. \r\n" );
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document.write( "That's because +4i is really 0+4i and so its conjugate is 0-4i,\r\n" );
document.write( "which is just -4i.\r\n" );
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document.write( "So we work the problem backwards:\r\n" );
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document.write( "We are to end up with:\r\n" );
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document.write( "    x = -3;        x = 4i         x = -4i\r\n" );
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document.write( "Before that we must have had this:\r\n" );
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document.write( "x + 3 = 0;    x - 4i = 0;    x + 4i = 0\r\n" );
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document.write( "Before that we must have had this:\r\n" );
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document.write( "            (x + 3)(x - 4i)(x + 4i) = 0\r\n" );
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document.write( "Multiplying the second and third parentheses together\r\n" );
document.write( "using FOIL:\r\n" );
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document.write( "     (x + 3)(x² + 4ix - 4ix - 16i²) = 0\r\n" );
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document.write( "Canceling the +4ix and the -4ix\r\n" );
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document.write( "                 (x + 3)(x² - 16i²) = 0\r\n" );
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document.write( "Replacing i² by -1\r\n" );
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document.write( "                (x + 3)[x² -16(-1)] = 0  \r\n" );
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document.write( "Simplifying:\r\n" );
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document.write( "                   (x + 3)[x² + 16] = 0\r\n" );
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document.write( "                   (x + 3)(x² + 16) = 0  \r\n" );
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document.write( "Multiplying those using FOIL:\r\n" );
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document.write( "                x³ + 16x + 3x² + 48 = 0\r\n" );
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document.write( "Arranging in descending order:\r\n" );
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document.write( "                x³ + 3x² + 16x + 48 = 0  \r\n" );
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document.write( "Edwin
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