Question 1207312
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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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<pre>
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) = {{{(-p/3)^3}}} + {{{p*(-p/3)}}} + d = {{{-p^3/27}}} - {{{p^2/3}}} + d;

hence,  d = {{{p^3/27}}} + {{{p^2/3}}}.


It is the <U>ANSWER</U>  to the problem's question.
</pre>

Solved.