document.write( "Question 656688: if alpha beta and gama are the root of the equation x^3-px+q =0then prove that alpha+beta+gama=0 \n" ); document.write( "
Algebra.Com's Answer #409600 by Edwin McCravy(20060)\"\" \"About 
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document.write( "\"x%5E3-px%2Bq+=+0\"\r\n" );
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document.write( "Since polynomial equations of the same degree with leading \r\n" );
document.write( "coefficient 1 (known as \"monic\" polynomial equations) and with \r\n" );
document.write( "the same roots and their multiplicities are unique, we create the \r\n" );
document.write( "cubic polynomial with leading coefficient 1, which will be the same\r\n" );
document.write( "as \"x%5E3-px%2Bq+=+0\"\r\n" );
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document.write( "\"x=alpha\", \"x=beta\", \"x=gamma\"\r\n" );
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document.write( "\"x-alpha=0\", \"x-beta=0\", \"x-gamma=0\"\r\n" );
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document.write( "\"%28x-alpha%29%28x-beta%29%28x-gamma%29=0\"\r\n" );
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document.write( "\"%28x%5E2-alpha%2Ax-beta%2Ax%2Balpha%2Abeta%29%28x-gamma%29=0\" \r\n" );
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document.write( "\"%28x%5E2-%28alpha%2Bbeta%29x%2Balpha%2Abeta%29%28x-gamma%29=0\" \r\n" );
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document.write( "This is the same polynomial equation as\r\n" );
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document.write( "\"x%5E3-px%2Bq+=+0\"\r\n" );
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document.write( "which is the same as\r\n" );
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document.write( "\"x%5E3%2B0x%5E2-px%2Bq+=+0\"\r\n" );
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document.write( "Equating coefficients of x²,\r\n" );
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document.write( "\"-%28alpha%2Bbeta%2Bgamma%29=0\"\r\n" );
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document.write( "\"alpha%2Bbeta%2Bgamma=0\"\r\n" );
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document.write( "Edwin
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