I am given the following sum:
$$\frac{x}{2x+1} + \frac{3}{x^2} + \frac{1}{x}$$
In order to add these fractions, I must find a common denominator. I have been taught to factor each denominator and then multiply each factor the greatest number of times that they occur. $$ (2x+1), x(x), x\\ LCD = (2x+1) \cdot x = 2x^2 + x $$ I have also been taught that you do not include anything that has been factored out (in this case, the second $x$ in $x(x)$). Thus, the greatest number of times $x$ occurs is once; similarly with $2x+1$, so the $LCD$ is the product of these two expressions.
But apparently this is not correct, and somehow, I have encountered multiple different solutions, including: $$ 2x^2 + x^3 \\ 2x^3 + x $$ What is correct? And please explain (simply) how I can calculate the $LCD$ for future problems like this.
Factoring each denominator and then multiplying each factor the greatest number of times that it occurs works. In this case, the factor $x$ occurs twice in the middle term, so the answer for least common denominator is $(2x+1)x^2=2x^3+x^2.$
The LCD has to be a multiple of all of the denominators, so it has to include all of the factors in the denominators for the greatest numbers of times they occur.