Question

algebra: concepts and connections - plc adding and subtracting rational expressions (continued) which shows the rational expression written using the least common denominator? \\(\frac{x - 2}{x^3} + \frac{x + 1}{2x^3}\\) \\(\frac{2(x - 2)}{x^3} + \frac{x + 1}{2x^3}\\) \\(\frac{(x - 2)}{2x^3} + \frac{x + 1}{2x^3}\\) \\(\frac{2(x - 2)}{2x^3} + \frac{x + 1}{2x^3}\\) \\(\frac{x - 2}{x^3} + \frac{x + 1}{x^3}\\)

Explanation:

Step1: Identify denominators

Denominators: $x^3$, $2x^3$

Step2: Find least common denominator

The least common multiple of $x^3$ and $2x^3$ is $2x^3$.

Step3: Rewrite first fraction to match LCD

Multiply numerator and denominator of $\frac{x-2}{x^3}$ by 2:
$\frac{(x-2) \times 2}{x^3 \times 2} = \frac{2(x-2)}{2x^3}$

Step4: Rewrite the expression

Keep the second fraction as is, since its denominator is already the LCD.

Answer:

$\boldsymbol{\frac{2(x-2)}{2x^3} + \frac{x+1}{2x^3}}$ (the top-left option)