When does the product of two polynomials =xk?Suppose f and g are are two polynomials with complex coefficents (i.e f,g∈C[x]). Let m be the order of f and let n be the order of g.Are there some general conditions where fg=αxn+m for some non-zero α∈C.

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sexec5m · Accepted Answer

Polynomials over C (in fact, over any field) are a Unique Factorization DomainSince x is an irreducible, the only way for that to happen is for f=axm and g=bxn, with ab=α.look at the lowest nonzero term in f and the lowest nonzero term in g; their product will be the lowest nonzero term in fg, hence must be of degree m+n. Since the degree of the lowest nonzero term of f is at most m and the one of g is at most n, you have that they must be exactly of degree m and n, respectively, and you get the result.

PCCNQN4XKhjx · Accepted Answer

We don&#039;t need the strong property of UFD. If D is a domain D then x is &#039; in D[x] (by&amp;nbsp;rmDxx=∼D&amp;nbsp;a domain), and products of &#039;s factor uniquely in every domain (same simple proof as in&amp;nbsp;Z). In particular, the only factorizations of the &#039; power&amp;nbsp;xi&amp;nbsp;are&amp;nbsp;rm,xjxk,&amp;nbsp;i=j+k&amp;nbsp;(up to associates as usual). This fails over non-domains, e.g.&amp;nbsp; x=(2x+3)(3x+2)∈ℤ/6[x].

When does the product of two polynomials

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