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Rational expressions are fractions wherein both the numerator and the denominator are polynomials. Just like fractions in basic arithmetic, rational expressions can be simplified, added, subtracted, multiplied, and divided. An important aspect of working with rational expressions is to find an equivalent expression that is easier to work with or to understand. This is where partial fraction decomposition comes into play.

Partial fraction decomposition is a process used to break down complex rational expressions into simpler ones, especially when the denominator contains multiple factors. This procedure facilitates the integration, differentiation, or simplification of algebraic expressions and is essential in calculus and higher algebra.

A prime quadratic factor refers to a quadratic polynomial that cannot be broken down into simpler linear factors. In other words, it's irreducible over the field of real numbers. The expression \( x^2+1 \) exemplifies a prime quadratic as it lacks real roots, and therefore it cannot be factored into real linear terms. When faced with a prime quadratic factor in the denominator of a rational expression, the partial fraction decomposition will include a term with the prime quadratic in the denominator and a linear expression in the numerator.

Factoring the denominator is a crucial step in partial fraction decomposition. It involves breaking down a polynomial into a product of its factors, which can be primes or higher-order polynomials. In many cases, factoring reveals repetitive or distinct linear factors that help to form the basis of the partial fraction decomposition setup.

For instance, a polynomial like \( x^3 - x^2 - x + 1 \) might factor into \( (x - 1)(x^2 + 1) \), containing both a linear factor \( x - 1 \) and a prime quadratic factor \( x^2 + 1 \) in its decomposition. It's important to remember that for proper partial fraction decomposition, the degree of the numerator should always be less than the degree of the denominator. If it's not, polynomial long division must be used first to rewrite the expression accordingly.

In partial fraction decomposition, solving for constants such as \( A, B, C, \) and \( D \), is where the algebra comes into play. Once you’ve established the form of the decomposition, the original rational expression equates to the sum of its partial fractions. By finding a common denominator and equating the numerators of both sides, you create an equation concerning your constants.

By matching coefficients or substituting convenient values for \( x \) to cancel out terms, we can solve for these unknowns. For example, if we set \( x \) to zero or any other value that simplifies the equation, this allows us to isolate and solve for one constant at a time. Once all constants are determined, they can be plugged back into the decomposed form to verify the solution.