Add the rational expressions to find the sum in simplest form: $\frac{x}{x-1}+\frac{x}{x+1}$.
We are tasked with simplifying the sum of two rational expressions: \(\frac{x}{x-1} + \frac{x}{x+1}\). To combine these expressions, we need to express them with a common denominator. The common denominator for the two rational expressions \(\frac{x}{x-1}\) and \(\frac{x}{x+1}\) is the product of their denominators: \((x-1)(x+1)\). According to the difference of squares identity, \((x-1)(x+1) = x^2 - 1\).
Next, we rewrite each fraction so that they share the common denominator \((x^2 - 1)\).
For the first term, we multiply the numerator and denominator by \((x+1)\):
\[\frac{x}{x-1} = \frac{x(x+1)}{(x-1)(x+1)} = \frac{x^2 + x}{x^2 - 1}\]
For the second term, we multiply the numerator and denominator by \((x-1)\):
\[\frac{x}{x+1} = \frac{x(x-1)}{(x+1)(x-1)} = \frac{x^2 - x}{x^2 - 1}\]
Now, we add the two fractions together by summing their numerators over the common denominator:
\[\frac{x^2 + x}{x^2 - 1} + \frac{x^2 - x}{x^2 - 1} = \frac{(x^2 + x) + (x^2 - x)}{x^2 - 1}\]
Grouping the terms in the numerator, we get:
\[\frac{x^2 + x^2 + x - x}{x^2 - 1} = \frac{2x^2}{x^2 - 1}\]
The simplified expression is \(\frac{2x^2}{x^2 - 1}\). Comparing this result with the provided options, we see that it matches the fourth choice.
Rational expressions appear frequently in physics and engineering when calculating total resistance in parallel circuits. If you have two resistors with resistances related to \(x-1\) and \(x+1\), the effective resistance might involve adding fractions like these to determine how the current distributes across the system. Simplifying such expressions helps engineers design more efficient electronic components by reducing the complexity of the equations involved.