I know for integers we have
$$\operatorname{lcm}(n,m) = \frac{nm}{\gcd(n,m)}$$
Does this hold for polynomials? i.e. $\operatorname{lcm}(f(x),g(x)) = \dfrac{f(x)g(x)}{\gcd(f(x),g(x))}$
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$\begingroup$ Do you mean lcm instead of lcd? $\endgroup$– ExtremalFeb 14, 2015 at 20:11
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1$\begingroup$ Polynomials over what type of coefficient ring? $\endgroup$– Bill DubuqueFeb 14, 2015 at 20:19
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$\begingroup$ Polynomials over a field $F_3$, that's all the information I have. And yes, I meant lcm, thanks Mathi! $\endgroup$– jstnchngFeb 14, 2015 at 20:46
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$\begingroup$ \gcd is a standard operator name; you don't need to write \operatorname{gcd}. ${}\qquad{}$ $\endgroup$– Michael HardyFeb 14, 2015 at 23:40
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1 Answer
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Yes it does hold but with a little difference! $\operatorname{L.C.M.}=\frac{P(x)\cdot Q(x)}{\operatorname{G.C.D.}(P(X),Q(X))\cdot \operatorname{DOM}(Q)\cdot\operatorname{DOM}(P)}$! $\operatorname{DOM}(P)$ is the leading coefficient! If $P(x)=1+x+3x^3$ hence $\operatorname{DOM}\{P\}=3$.
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$\begingroup$ Hmm. So if I'm looking for the lcm of f(x) = $x^3-x^2-1$ and g(x) = $x^2-x+1$, I can just take f(x)g(x)/ gcd(f(x),g(x))? $\endgroup$– jstnchngFeb 14, 2015 at 20:15
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