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.

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(52800) (Show Source): You can put this solution on YOUR website!
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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.
~~~~~~~~~~~~~~~~~~~~

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.

Solved.

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