document.write( "Question 155177: a,b,c are three distinct real numbers and there are real numbers x,y such that a^3+ax+y=0,b^3+bx=y=0 and c^3+cx+y=0 Show that a+b+c=0 \n" ); document.write( "
Algebra.Com's Answer #115356 by Edwin McCravy(20054)\"\" \"About 
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a,b,c are three distinct real numbers and there are real numbers x,y such that a3+ax+y=0,b3+bx=y=0 and c3+cx+y=0 Show that a+b+c=0\r
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document.write( "All three equations are cases of the cubic \r\n" );
document.write( "polynomial equation:\r\n" );
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document.write( "z3+xz+y=0\r\n" );
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document.write( "Since a3+ax+y=0,b3+bx=y=0 and c3+cx+y=0, this shows\r\n" );
document.write( "that z=a, z=b and z=c are distinct solutions to \r\n" );
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document.write( "z3+xz+y=0\r\n" );
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document.write( "and since that equation is of degree 3 it can have no\r\n" );
document.write( "more than 3 solutions. Thus z=a, z=b, and z=c are the \r\n" );
document.write( "three real solutions to\r\n" );
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document.write( "z3+xz+y=0\r\n" );
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document.write( "Therefore its left side must factor as\r\n" );
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document.write( "(z-a)(z-b)(z-c)=0\r\n" );
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document.write( "(z-a)(z2-cz-bz+bc)=0\r\n" );
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document.write( "z3-cz2-bz2+bcz-az2+acz+abz-abc=0\r\n" );
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document.write( "z3-(a+b+c)z3+(ab+ac+bc)z-abc=0\r\n" );
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document.write( "This must be the same equation as\r\n" );
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document.write( "z3+xz+y=0\r\n" );
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document.write( "Therefore the coefficient of z2 in\r\n" );
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document.write( "z3-(a+b+c)z2+(ab+ac+bc)z-abc=0\r\n" );
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document.write( "must be 0 since z3+xz+y=0 has no\r\n" );
document.write( "term in z2.  Therefore\r\n" );
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document.write( "-(a+b+c)=0 or\r\n" );
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document.write( "a+b+c=0.\r\n" );
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document.write( "That's all you were asked to show.\r\n" );
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document.write( "However by equating the other terms,\r\n" );
document.write( "you can also find x and y:\r\n" );
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document.write( "x=ab+ac+bc and y=-abc.\r\n" );
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document.write( "Edwin

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