When does the produt 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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When does the produt 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

einfachmoipf · Accepted Answer

Step 1 We don&#039;t need the strong property of UFD. If D is a domain D then x is &#039; in D[x] (by D[x]x=∼D a domain), and products of &#039;s factor uniquely in every domain (same simple proof as in Z). In particular, the only factorizations of the &#039; power xi are xjxk, i=j+k (up to associates as usual). This fails over non-domains, e.g. x=(2x+3)(3x+2)∈Z6[x]

Juan Spiller · Accepted Answer

Step 1  The intuitively evident ones: all other terms in f and g must vanish.  To see this, note that the product of the constant terms of f and g equals the constant term of fg, which is zero, whence at least one of these polynomials is multiple of x.  Without any loss of generality assume it is f. Then  fg=x(fx)g,  implying (fx)g is a multiple of xn+m−1.  By induction this reduces us to the case  n+m=0,  which is trivial (because f and g then have no other terms). QED.

alenahelenash · Accepted Answer

Polynomials over C (in fact, over any field) are a Unique Factorization Domain since x is an irreducible, the only way for that to happen is forf=axmandg=bxn,with ab=α.If you don&#039;t want to bring in the sledgehammer of unique factorization, you can just do it explicitly: 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

When does the produt of two polynomials =x^{k}?Suppose f and

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