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A rational function is a fraction in which both the numerator and the denominator are polynomials. These functions are common in algebra and calculus because they represent the division of two polynomial expressions. Understanding rational functions is essential to comprehend different mathematical concepts such as asymptotes, holes, and limits, as these features often arise from the behavior of rational functions.

In the given exercise, the rational function is \( \frac{3x+1}{(x-1)^2} \). It is important to note that the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator. When dealing with rational functions, this property tells us that we can apply partial fraction decomposition to simplify the expression and analyze it more easily.

As part of simplifying rational functions, partial fraction decomposition allows us to express a complex fraction as a sum of simpler fractions. This technique is particularly useful in calculus when dealing with integrals of rational functions, providing a way to integrate these expressions by breaking them into more manageable parts.

Coefficients are the numerical factors in terms of polynomials. In the context of partial fraction decomposition, these coefficients are crucial because they help determine the specific values that accompany each term of the simplified fractions.

In our exercise, once we decompose the given rational function, we end up with an equation: \( \frac{3x+1}{(x-1)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2} \). The letters \(A\) and \(B\) are the coefficients we need to find to complete the decomposition.

To determine these coefficients, we multiply through by the common denominator and compare coefficients on each side of the equation. The steps involve:

• Expanding and arranging the resulting polynomial equations.
• Matching the coefficients of like terms from both sides of the equation.

For this function, the comparison yields the equations:

• For the \(x\) term, we have \(A = 3\).
• For the constant term, \(-A + B = 1\), which resolves to \(B = 4\) after substituting \(A = 3\).

Finding these coefficients accurately is vital to constructing the reconstructed simpler rational functions.

The structure of the denominator in rational functions is perhaps one of the most important aspects to examine when performing partial fraction decomposition. It affects how the function gets broken down into simpler fractions.

For the decomposition process, we carefully analyze the denominator, which, in this exercise, is \((x-1)^2\). The key takeaway from this is recognizing the repeated, or multiplicity, factor within the denominator.

The general process when handling denominators involves assessing:

• Repeated linear factors, as seen in our example.
• Distinct linear factors, which each result in an individual term.
• Potential quadratic or higher-degree irreducible factors.

The repeated factor \((x-1)\) results in terms of \(\frac{A}{x-1}\) and \(\frac{B}{(x-1)^2}\) in the decomposition.This approach stems from ensuring each potential divisor is appropriately represented in the decomposition.

Understanding the structure of the denominator thoroughly helps ensure successful and correct partial fraction decomposition, aiding in the simplification and integration of complex rational functions.