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

Question

Eva Adamson · Accepted Answer

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.
#4w+2 = 2(2w+1)#
#4w^2-1 = (2w+1)(2w-1)# So the unique factors are: #3#, #2#, #(2w+1)# and #(2w-1)#. 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*2*(2w+1)(2w-1) = 6*(4w^2-1) = 24w^2-6#

Isaiah Anthony · Answer

undefined To find the least common multiple (LCM) for 3, 4w+2, and 4w^2-1, we need to factorize each expression and then determine the highest power of each factor that appears in any of the expressions.

The prime factorization of 3 is 3.
The prime factorization of 4w+2 is 2(2w+1).
The prime factorization of 4w^2-1 is (2w+1)(2w-1).

Now, we consider the highest power of each factor:
- The highest power of 2 is 2.
- The highest power of 3 is 1.
- The highest power of (2w+1) is 1.
- The highest power of (2w-1) is 1.

Multiplying these highest powers together, we get:
2^2 * 3^1 * (2w+1)^1 * (2w-1)^1

Simplifying this expression, the LCM for 3, 4w+2, and 4w^2-1 is:
12(2w+1)(2w-1)