Question

Express the rational function as a sum or difference of two simpler rational expressions. \(\frac{2 x}{(x+2)^{2}}\)

Short answer

The expression is \( \frac{1}{x+2} + \frac{-1}{x+2} + \frac{2}{(x+2)^2} \).

Step-by-step solution

Decompose the function

To express the rational expression \( \frac{2x}{(x+2)^2} \) as a sum or difference of two simpler fractions, consider trying to split the numerator in a way that allows for partial fraction decomposition.

Rewrite numerator

Notice that the numerator \(2x\) can be rewritten as: \( (x+2) - (x-2) \). We use this decomposition to split the fraction. So, \( \frac{2x}{(x+2)^2} \) can be rewritten as \( \frac{x+2}{(x+2)^2} + \frac{-(x-2)}{(x+2)^2} \).

Simplify expressions

Simplify each of the fractions: \( \frac{x+2}{(x+2)^2} = \frac{1}{x+2} \) and \( \frac{-(x-2)}{(x+2)^2} = \frac{-1}{x+2} + \frac{2}{(x+2)^2} \).

Combine fractions

Combine the results from the previous step: \( \frac{2x}{(x+2)^2} = \frac{1}{x+2} + \left( \frac{-1}{x+2} + \frac{2}{(x+2)^2} \right) = \frac{2}{(x+2)^2} \).

Key concepts

Rational Expressions

At the heart of algebraic manipulation lies the concept of rational expressions. Simply put, rational expressions are fractions where both the numerator and the denominator are polynomials. They are quite similar to numerical fractions but involve variables instead of just numbers. For example, in the expression \( \frac{2x}{(x+2)^2} \), the numerator is \( 2x \) and the denominator is \( (x+2)^2 \).

Understanding rational expressions is crucial as they often appear in equations and other functions that need to be solved or simplified. Handling them correctly can help in solving algebraic equations efficiently. Key skills include finding common denominators and simplifying the expression by canceling common factors.

Remember:

• Numerator: The top part of a fraction.
• Denominator: The bottom part of a fraction.
• Always ensure the denominator is not zero to avoid mathematical errors.

Numerator Decomposition

Numerator decomposition is a strategic method used to simplify complex rational expressions. It involves breaking down the numerator into simpler parts that can be managed more easily. This approach is pivotal for partial fraction decomposition, a technique used to express a single complex rational expression as a sum or difference of simpler fractions.

In the example \( \frac{2x}{(x+2)^2} \), the numerator \( 2x \) is rewritten as \( (x+2) - (x-2) \). This decomposition allows us to split the complex fraction into simpler components: \( \frac{x+2}{(x+2)^2} \) and \( \frac{-(x-2)}{(x+2)^2} \).

By doing so, it becomes easier to manage and solve the expression by applying further simplification techniques. Always look for ways to express the numerator as a combination of terms that match or relate to parts of the denominator, as this can simplify the entire fraction handling process.

Tips for effective decomposition:

• Identify relationships between numerator and denominator terms.
• Break down the numerator into terms that can be individually divided by the denominator.
• Ensure the decomposition allows for simplification without changing the original expression value.

Simplification of Fractions

Simplification is the process of making a fraction or an expression as simple as possible, by reducing its complexity. When dealing with rational expressions, simplification often involves canceling out common terms or factors from the numerator and the denominator.

In the illustrated solution, the final simplification step involved breaking down \( \frac{2x}{(x+2)^2} \) into simpler fractions. By rewriting it as \( \frac{x+2}{(x+2)^2} + \frac{-(x-2)}{(x+2)^2} \), and then reducing these terms, we can simplify the expression to \( \frac{2}{(x+2)^2} \).

For successful simplification:

• Always reduce the greatest common factor from both numerator and denominator, if possible.
• Explore the factorization of polynomials to reveal simplification opportunities.
• Ensure the operation respects the mathematical equivalence of expressions.
In any simplification process, the goal is to express the fraction in its simplest, most compact form while maintaining its original value and properties.