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Rational expressions are fractions that contain polynomials in both the numerator and the denominator. They are the ratio of two polynomials, much like a ratio of two numbers. Understanding how to handle such expressions is crucial in algebra, as it allows for the simplification of complex algebraic fractions and can be vital in calculus when integrating rational functions.

A key process in working with rational expressions is partial fraction decomposition. This is a method used to break apart rational expressions into simpler, more manageable pieces, especially when the denominator is complex. The expression \(\frac{4 x^{2}+13 x-9}{x(x-1)(x+3)}\) from the given exercise is a canonical example.

When approaching rational expressions, the goal is often to rewrite them in a form that is easier to work with, especially when integrating or finding limits. Thus, being proficient with partial fraction decomposition is a highly valuable skill when studying higher levels of mathematics.

Linear factors refer to expressions of the first degree, which means they are polynomials of the highest exponent of 1. The fundamental theorem of algebra states that every non-zero single-variable polynomial with complex coefficients has as many complex roots as its degree, implying it can be factored into linear factors over the complex numbers. When it comes to real numbers, which is more common in introductory algebra courses, a polynomial is factored into linear (if all roots are real) and irreducible quadratic factors (if it has complex roots).

In the context of partial fraction decomposition, we find that a rational expression's denominator can often be factored into linear factors, such as \(x\), \(x-1\), and \(x+3\) in our exercise. Each of these factors can contain a different variable, an integer, or both. Recognizing and factoring a polynomial into linear factors is essential for partial fraction decomposition as it allows us to express the rational expression as a sum of fractions, each with a linear factor in the denominator.

Equating coefficients is a method used in algebra to determine the values of unknown coefficients in polynomial expressions. When two polynomials are equal, their corresponding coefficients must be equal as well. As employed in partial fraction decomposition, after you have distributed the presumed A, B, and C throughout the factors, you need to compare coefficients from both sides of the resultant equation.

For example, in our exercise, after clearing the fractions we compared the coefficients of like terms on both sides to set up the system of equations. By doing so, we effectively equated the coefficients of \(x^2\), \(x\), and the constants from the expanded expression with those of the original rational expression. This comparison gave us a set of linear equations that we then solved to find the values of A, B, and C. Mastery of this technique is vital for students as it underpins many algebraic processes and is widely utilized in calculus, differential equations, and beyond.