document.write( "Question 1057047: Find a polynomial equation with real coefficients that has
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Algebra.Com's Answer #672143 by Edwin McCravy(20054) $\"\"$ $\"About$

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Find a polynomial equation with real coefficients that has
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document.write( "Since 2+i is a solution, and since the coefficients are all real,\r\n" );
document.write( "its conjugate 2-i is also a solution.  Begin with\r\n" );
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document.write( "     x = 5;  x = 2+i;  x = 2-i\r\n" );
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document.write( "Get 0 alone on the right side of each:\r\n" );
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document.write( "   x+5 = 0;  x-2-i = 0; x-2+i = 0\r\n" );
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document.write( "Multiply all left sides together [and right sides too! (0)(0)(0)=0]\r\n" );
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document.write( "      (x+5)(x-2-i)(x-2+i) = 0\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "  (x+5)[(x-2)-i][(x-2)+i] = 0\r\n" );
document.write( "         (x+5)[(x-2)²-i²] = 0\r\n" );
document.write( "       (x+5)[(x-2)²-(-1)] = 0\r\n" );
document.write( "          (x+5)[(x-2)²+1] = 0\r\n" );
document.write( "      (x+5)[(x-2)(x-2)+1] = 0\r\n" );
document.write( "         (x+5)[x²-4x+4+1] = 0\r\n" );
document.write( "           (x+5)(x²-4x+5) = 0\r\n" );
document.write( "             x³+x²-15x+25 = 0\r\n" );
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document.write( "So those zeros would be obtained when we set the\r\n" );
document.write( "function f(x) = x³+x²-15x+25 equal to 0.  So\r\n" );
document.write( "one answer is: \r\n" );
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document.write( "f(x) = x³+x²-15x-25\r\n" );
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document.write( "Other answers would be gotten by multiplying the\r\n" );
document.write( "right side by different constants k:\r\n" );
document.write( "f(x) = kx³+kx²+25kx-15k\r\n" );
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document.write( "Edwin

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