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A rational expression is simply a fraction in which both the numerator and denominator are polynomials. Polynomials are algebraic expressions made up of variables and constants connected by addition, subtraction, and multiplication. In the realm of rational expressions, these can be as simple as linear equations or as complex as quadratics and higher-degree polynomials.

The rational expression given, \( \frac{-x^{2}+15}{4x^{4}+40x^{2}+36} \), requires us to manipulate polynomials in both the numerator and the denominator. Our task involves finding a way to break down this complex expression into simpler components through partial fraction decomposition. This is a technique used to rewrite the expression so each part is easier to analyze and integrate if needed.

Understanding the structure and behavior of rational expressions is crucial for solving many algebraic problems. It allows us to perform operations such as addition, subtraction, and integration on complex polynomial expressions.

Factoring is a mathematical process of breaking down a complex expression into simpler parts, called factors, which when multiplied together will yield the original expression. It's like solving a puzzle where the pieces come together to form the bigger picture.

For the denominator \(4x^4 + 40x^2 + 36\), factoring helps us identify repetitive components or simpler expressions within a complex polynomial. To start factoring, note that it takes the form of a quadratic in terms of \(x^2\), which can be written as \(4(x^2)^2 + 40x^2 + 36\). By letting \(u = x^2\), the expression simplifies to \(4u^2 + 40u + 36\).

Factoring the quadratic \(u\) gives us \((2u + 6)(2u + 6)\), which eventually results in \((2(x^2 + 3))^2\) when plugged back. This step is crucial for simplifying the decomposition. Factoring simplifies the expression for use in partial fraction decomposition, making it easier to work with later on.

In a rational expression, the denominator is the polynomial that sits beneath the fraction bar. It's important because it determines the nature and constraints of the rational expression.

For the expression \(\frac{-x^2 + 15}{4x^4 + 40x^2 + 36}\), the denominator \(4x^4 + 40x^2 + 36\) underwent a transformation through factoring to become \((2(x^2 + 3))^2\). This is a more manageable form revealing that the denominator has a repeated factor. Such repetition directly impacts the form of the partial fraction decomposition.

When dealing with rational expressions, the denominator plays a crucial role in finding limits, asymptotes, and discontinuities. Incorrect handling of the denominator can lead to mistakes in the solution process, particularly in partial fraction decomposition and integration.

Coefficients are the numerical or constant parts of a term that are multiplied by the variable(s) in an algebraic expression. They are fundamental in algebraic manipulation and form the essence of solving equations.

In the context of the given partial fraction decomposition solution, coefficients were identified by expanding the expression and matching terms with corresponding powers of \(x\) in the untouched polynomial. For example, identifying zero coefficients for\(x^3\) and \(x\) helped ensure accuracy in matching it with the expression \(-x^2 + 15\).

Determining each coefficient ensures that the decomposition accurately reflects the original expression. It requires solving simultaneous equations, often employing strategies like comparison of coefficients, resulting in terms like \(2B = -1\) and \(6B + D = 15\).

This step highlights how each coefficient contributes to the overall form of the partial fraction. The proper identification and calculation of coefficients are vital to successfully completing partial fraction decomposition.