Final answer:
The least common multiple of 3w^2-75 and w-5 is 3(w - 5)(w + 5). This result comes from factoring both expressions and taking the product of the highest powers of all factors present.
Step-by-step explanation:
To find the least common multiple (LCM) of 3w^2-75 and w-5, we want to express each term as a product of their factors. First, we can factor out the greatest common factor (GCF) of 3w^2-75, which is 3. This yields:
3(w^2 - 25)
The quadratic term w^2 - 25 is a difference of squares and can be factored further into:
3(w - 5)(w + 5)
Since w - 5 is already in its factored form and is a factor of 3w^2-75, the LCM of these two expressions is simply:
3(w - 5)(w + 5)
This is because LCM of two algebraic expressions is the product of the highest powers of all factors involved. In this case, w - 5 appears in both expressions and w + 5 is an additional factor from the first expression, the second expression does not contribute any new factors.