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Understanding the concept of a rational expression is vital when dealing with algebraic fractions and their decompositions. A rational expression is a quotient of two polynomials, much like a fraction where you have a numerator and a denominator, except that both of these parts are polynomials. For example, in the expression \(\frac{x}{(x+1)^2}\), \(x\) is the polynomial numerator and \((x+1)^2\) is the polynomial denominator.

Why is it important to grasp what a rational expression is? When you're simplifying, adding, subtracting, multiplying, or dividing algebraic fractions, you need to understand their structure as rational expressions to perform the operations correctly. Furthermore, the process of breaking down these expressions into simpler parts – that's where partial fraction decomposition comes in – is a fundamental skill in higher-level calculus. It's widely used in fields such as engineering, physics, and economics for simplifying complex rational expressions to make calculus techniques, like integration, more manageable.

In partial fraction decomposition, recognizing repeated linear factors is crucial. A linear factor is a product term that consists of a variable raised to the first power, usually represented as \((x-n)\), where \(n\) is a constant. When the same linear factor occurs more than once – in other words, it's repeated as in \((x-n)^k\) where \(k\) is an integer greater than 1 – we say we have repeated linear factors.

Take the given exercise example \(\frac{x}{(x+1)^2}\). Here, \((x+1)\) is a linear factor that is squared, indicating it's repeated. This repetition changes how we approach the decomposition. For every repeated factor, you'll have several terms in the partial fractions, incrementing the power of the denominator from 1 up to \(k\), which in this case is 2. It means we need to account for \((x+1)\) and \((x+1)^2\) separately in our decomposition because each represents a unique piece of the puzzle that, once solved, gives us more insight into the behavior of the original rational expression.

In the context of partial fraction decomposition, solving for coefficients is a method we use to determine the values that make up the simpler fractions that comprise our original rational expression. The coefficients are typically represented by the capital letters \(A, B, C,\) etc., and they are crucial in reconstructing the original rational expression from its decomposed parts.

The process of finding these coefficients involves setting up an equation based on the proposed decomposition, then determining the unknowns. Insights from algebra are used, such as equating coefficients from both sides of an equation, to find the specific values of \(A, B, C,\) etc. In the step-by-step solution given, we see this in action: comparing coefficients on both sides of the equation \(x = A(x + 1) + B\) to uncover the values of \(A\) and \(B\). Once these coefficients are known, the partially decomposed fractions can be written out fully, revealing a more straightforward structure than that of the original expression.