Final answer:
The LCD for the expressions (2/5c^2 + 39c + 54) and (3c/5c^2 - C - 18) should be the product of the distinct polynomial factors. In this case, it would be 5c^2(5c^2 - C - 18), but this does not match any of the provided options. The closest given option by structure is 5c^2 - C - 18.
Step-by-step explanation:
To find the LCD (Least Common Denominator) for the given expressions (2/5c^2 + 39c + 54) and (3c/5c^2 - C - 18), we must identify the denominators and then determine the smallest expression that both denominators can divide into without a remainder.
Looking at the denominators 5c^2 and 5c^2 - C - 18, we see that both have the term 5c^2, but the second has additional terms. To find the LCD, we need to consider both of these terms.
There is no need to factor the denominators since one is a simple monomial and the other is a trinomial that does not share a factor with the first. In this case, the LCD is the product of the distinct polynomial factors, which is simply 5c^2(5c^2 - C - 18). However, none of the given options represents this product, suggesting a typo or mistake in the provided options. If we assume it to be an oversight, then closest to our LCD by structure is option (a) 5c^2 - C - 18.