document.write( "Question 1207312: Given that f(x) = x ^ 3 + px + d and g(x) = 3x ^ 2 + px have a common factor, where p and d are non-zero constants, find the relation between pand d. \n" ); document.write( "

Algebra.Com's Answer #845115 by ikleyn(52798) $\"\"$ $\"About$

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\n" ); document.write( "Given that f(x) = x ^ 3 + px + d and g(x) = 3x ^ 2 + px have a common factor,
\n" ); document.write( "where p and d are non-zero constants, find the relation between p and d.
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document.write( "Saying that the given polynomials, f(x) = x^3 + px + d and g(x) = 3x^2 + px\r\n" );
document.write( "have a common factor MEANS that they have a common LINEAR factor.\r\n" );
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document.write( "In turn, it means that these polynomials have a common root (at least one).\r\n" );
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document.write( "The polynomial g(x) = 3x^2+px = x*(3x+p) has the roots x=0 and x= -p/3.\r\n" );
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document.write( "Since d =/= 0 (given !), it means the x= 0  IS NOT  a root to f(x).\r\n" );
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document.write( "Thus f(x) has the root x= -p/3.\r\n" );
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document.write( "Then  0 = f(-p/3) =  +  + d =  -  + d;\r\n" );
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document.write( "hence,  d =  + .\r\n" );
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document.write( "It is the ANSWER  to the problem's question.\r\n" );
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