91Ó°ÊÓ

When dealing with rational expressions, much like with fractions, finding a common denominator is crucial for adding or subtracting them.
For example, in general fractions such as \(\frac{1}{2} + \frac{1}{3}\), the denominators are different—2 and 3. To combine them, we must find a common multiple of these denominators.
The same idea applies to rational expressions that involve variables.

Imagine you have the rational expressions \(\frac{2x}{x-1}\) and \(\frac{3}{x}\). Here, the denominators are \(x-1\) and \(x\).
To add these expressions, we determine the least common denominator (LCD).

• The LCD is the smallest expression that both \(x-1\) and \(x\) can divide into, without leaving a remainder.
• In this case, it is simply their product: \((x-1) \cdot x\).

Understanding how to find the common denominator helps pave the way to rewriting each expression in a comparable form. This is necessary to add or subtract them later on.

Simplifying expressions is a key step in the process of working with rational expressions.
After finding a common denominator, the next step involves rewriting each fraction to reflect this common denominator.

For example, if you have \(\frac{2x}{x-1}\) and \(\frac{3}{x}\), the LCD is \((x-1)x\).
Thus, you would adjust each expression accordingly to match this.

• For \(\frac{2x}{x-1}\): Multiply both the numerator and denominator by \(x\), resulting in \(\frac{2x^2}{x(x-1)}\).
• For \(\frac{3}{x}\): Multiply both the numerator and denominator by \(x-1\), resulting in \(\frac{3(x-1)}{x(x-1)} = \frac{3x-3}{x(x-1)}\).

Once each fraction shares the same denominator, they can be added or subtracted and potentially further simplified once combined.
Remember: simplifying is about clarity and minimizing complexity.

• This often means combining like terms and trying to factor expressions where possible. In our case, further simplification was not possible beyond combining the terms under a single denominator.

Adding fractions, whether they're numerical or involve variables like in rational expressions, follows the same principle:
Once you have a common denominator, the addition occurs over that shared base.

In our example, the two modified fractions become \(\frac{2x^2}{x(x-1)}\) and \(\frac{3x-3}{x(x-1)}\).

• Since the denominators are now the same, the fractions can be combined by adding the numerators.
• Thus, you find \(\frac{2x^2 + (3x - 3)}{x(x-1)}\).

It is crucial to carefully handle the numerators during this process.
Add, subtract, multiply, or divide the coefficients as needed but keep an eye on variable terms, ensuring they are correctly combined.

For the rational expression in question, this results in a single, streamlined expression that has been simplified where possible: \[ \frac{2x^2 + 3x - 3}{x(x-1)} \].
Although this expression didn’t simplify further through factoring, it exemplifies the concept of merging through addition after achieving a common denominator.