SOLUTION: 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.
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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. Answer by ikleyn(52795) (Show Source):
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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 p and d.
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Saying that the given polynomials, f(x) = x^3 + px + d and g(x) = 3x^2 + px
have a common factor MEANS that they have a common LINEAR factor.
In turn, it means that these polynomials have a common root (at least one).
The polynomial g(x) = 3x^2+px = x*(3x+p) has the roots x=0 and x= -p/3.
Since d =/= 0 (given !), it means the x= 0 IS NOT a root to f(x).
Thus f(x) has the root x= -p/3.
Then 0 = f(-p/3) = + + d = - + d;
hence, d = + .
It is the ANSWER to the problem's question.
