Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.(11x - 10)/(x − 2) (x + 1)

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Identify the denominator factors of the rational expression. Here, the denominator is \( (x - 2)(x + 1) \), which consists of two distinct linear factors.

Since the denominator factors are linear and distinct, set up the partial fraction decomposition as a sum of fractions with unknown constants in the numerators over each linear factor:

\[ \frac{11x - 10}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \]

Here, \( A \) and \( B \) are constants that would be determined if solving the decomposition completely, but for this problem, only the form is required.

This form expresses the original rational expression as a sum of simpler rational expressions, each with a single linear denominator.

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주요 개념

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Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with linear or quadratic denominators. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factorization of Denominators

Understanding how to factor the denominator into linear or irreducible quadratic factors is essential for setting up the correct form of partial fractions. Each factor determines the structure of the terms in the decomposition, such as constants over linear factors or linear expressions over quadratic factors.

Form of Partial Fractions for Distinct Linear Factors

When the denominator consists of distinct linear factors, the partial fraction decomposition is written as a sum of fractions with unknown constants in the numerators over each linear factor. For example, for (x−2)(x+1), the form is A/(x−2) + B/(x+1), where A and B are constants to be determined.