Question

What is the LCM of `3m^3-24` and `m^2-4`?

Answer 1

`LCM=3(m-2)(m+2)(m^2+2m+m^2)`

Explanation:

Factorise the expressions first:

`3m^3 -24 = 3(m^3-8)" "larr` we now have difference of cubes

`=3color(blue)((m-2))(m^2+2m+m^2)" "larr` there are 3 factors

`m^2-4 = (m+2)color(blue)((m-2))" "larr` there are 2 factors

The LCM must be divisible by both expressions.

Therefore all the factors of both expressions must be in the LCM, but without any duplicates. There is a common factor in both expressions: `color(blue)((m-2))` is in both expressions, only one is needed in the LCM.

`LCM=3color(blue)((m-2))(m^2+2m+m^2) xx (m+2)`

`=3(m-2)(m+2)(m^2+2m+m^2)" "larr` there are 4 factors