# LCD of 2x+1, x^2 and x

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.

• Welcome to MSE. The LCD is $x^2(2x+1)=2x^3+x^2$ – J. W. Tanner Mar 26 at 22:21
• How do you calculate the LCD of 2 and 9? – dcolazin Mar 26 at 22:24
• or of $2, 9,$ and $3$ (to be more analogous)? – J. W. Tanner Mar 26 at 22:30
• "But apparently this is not correct, and somehow, I have encountered multiple different solutions, " Where did you encounter these? – fleablood Mar 26 at 22:49
• " I have encountered multiple different solutions, including: " And I've encountered people saying the earth is flat and Donald Trump isn't scum. The LCD of $2x+1, x\cdot x, x$ is $x\cdot x \cdot(2x+1) = x^2(2x+1)$ exactly as you thought for exactly the reason you said it was. Don't listen to idiots. – fleablood Mar 26 at 23:07

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 $$(2x+1)x^2=2x^3+x^2.$$