Express \(\frac{4}{x^{2}\left(x^{2}-4 x+4\right)}\) as the sum of fractions. a) \(\frac{1}{x}-\frac{1}{x-2}+\frac{1}{(x-2)^{2}}\) c) \(\frac{1}{x^{2}}+\frac{1}{(x-2)^{2}}\) b) \(\frac{1}{x}+\frac{1}{x^{2}}-\frac{1}{x-2}+\frac{1}{(x-2)^{2}}\) d) \(\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x-2}+\frac{1}{(x-2)^{2}}\)

Algebraic fractions are simply fractions that contain algebraic expressions, like variables and exponents, in their numerators, denominators, or both. Understanding algebraic fractions is essential for solving many types of math problems, especially when it comes to integration or finding limits in calculus.

Let's consider how algebraic fractions can be broken down into simpler parts through the process called partial fraction decomposition. This technique is particularly useful when dealing with integration. In the exercise, the algebraic fraction \(\frac{4}{x^{2}(x^{2}-4 x+4)}\) was decomposed into simpler fractions. This is a step-by-step process that involves finding a common denominator and then breaking the fraction down into a sum of simpler fractions.

The concept of a common denominator is crucial when working with algebraic fractions. A common denominator is the shared multiple of all the denominators of a set of fractions. It allows us to combine or compare fractions by converting them into equivalent fractions with the same denominator.

In our example, we looked for a common denominator when testing different options for the partial fraction decomposition. In the context of algebraic fractions, finding a common denominator can become more complex, as it often involves factoring polynomial expressions. For instance, we tested option D with all fractions having the common denominator of \(x^2(x-2)^2\). It's important to find the least common denominator that allows all the individual fractions to be expressed as equivalent fractions with this shared denominator. This is why in the solution steps, we always rewrote each option using a common denominator before combining them.

Rational expressions, like \(\frac{4}{x^{2}(x^{2}-4 x+4)}\), are fractions where both the numerator and the denominator are polynomials. They behave much like the rational numbers (fractions). Yet, unlike simple fractions, rational expressions can take on a wide array of characteristics depending on the nature of their polynomials.

In partial fraction decomposition, our goal is to express these complex rational expressions as a sum of simpler fractions, which can then be used for further mathematical processes such as integration. Evaluating our given exercise, we find that the rational expression is best expressed as the sum of the components listed in option D.

The process also involves simplification, where terms might be combined or canceled out, showcasing how versatile and intricate rational expressions can be. When you fully grasp the concept of rational expressions, concepts like partial fraction decomposition become much easier to tackle.