How to find the LCM of two monomials or polynomials

Question

How would I go about finding the LCM of two monomials, such as $240a^2bc,700a^3b^2$ , or polynomials such as $240a^bc-3, 700a^3b^2+5$? The only quick way I could think of would be to multiply them together, but that isn't always the least common multiple, it's just a multiple.

*note: I'm not looking for the answer to the problem, I'd like to know how to do it myself

Answer 1

Similarly to finding the LCM of two integers, where you factor them and take the maximum power of each prime, you take the maximum power of each variable. The LCM of $a^2bc$ and $a^3b^2$ is therefore $a^3b^2c$. You find the LCM of $240$ and $700$ the usual way and multiply it on the front. If your polynomials have multiple terms, you need to factor them as completely as possible, then apply this approach. So, for example, the LCM of $x^3-3x+2=(x-1)^2(x+2)$ and $x^4+5 x^3+6 x^2-4 x-8=(x+2)^3(x-1)$ is $(x-1)^2(x+2)^3$

Answer 2

Find the GCD of the polynomials, which can be done using the Euclid Algorithm and then use that

$$\frac{P \cdot Q}{GCD(P,Q)}$$

where $P,Q$ are some polynomials.