Question 1173277
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If you consider the ring of polynomials with real coefficients,  it is  NOT necessary


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    i.e., the first fact not necessary implies the second fact.
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If you consider the ring of polynomial with complex number coefficients, 

and if the mentioned roots of the polynomial &nbsp;f(x) &nbsp;all have multiplicities lesser than 

the corresponding roots of &nbsp;g(x), &nbsp;then it &nbsp;IS &nbsp;necessary: the first fact implies the second.



But if the condition about multiplicities is not held, &nbsp;then the first fact does not necessary implies the second fact.


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Example :   g(x) = (x-(-5)) * (x-(-3)) * (x-2) * (x-4) * (x-8)


            f(x) = (x-(-3))^2 * (x-2)^3 * (x-8)^9
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Over the complex domain, &nbsp;the necessary additional information is about &nbsp;MULTIPLICITIES &nbsp;of the roots.


If multiplicities of all corresponding/relevant roots of &nbsp;g(x) &nbsp;are greater than or equal to those of the polynomial &nbsp;f(x),

then the divisibility  &nbsp;g(x) = 0 &nbsp;&nbsp;( mod f(x) )  &nbsp;&nbsp;takes place;

if not - the divisibility does not take place.