How do you find the LCM for 3, 4w+2 and 4w^2-1?

Jun 8, 2015

One way is to factor each of them completely, then multiply the non-duplicated factors together. I say non-duplicated, but if a factor is repeated in one of the factorisations of the starting expressions, then it has to occur at least that many times in the LCM.

Anyway:

$3$ is completely factored already.
$4 w + 2 = 2 \left(2 w + 1\right)$
$4 {w}^{2} - 1 = \left(2 w + 1\right) \left(2 w - 1\right)$

So the unique factors are: $3$, $2$, $\left(2 w + 1\right)$ and $\left(2 w - 1\right)$.

None of these factors occurs in one of the original expressions with a multiplicity greater than $1$, so we can just multiply them together to get the LCM:

$3 \cdot 2 \cdot \left(2 w + 1\right) \left(2 w - 1\right) = 6 \cdot \left(4 {w}^{2} - 1\right) = 24 {w}^{2} - 6$