5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
6. (4x + 1)/(4x² - 1)

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Identify the denominator and factor it completely. The denominator is \(4x^{2} - 1\), which is a difference of squares and factors as \(\left(2x - 1\right)\left(2x + 1\right)\).

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

\[\frac{4x + 1}{(2x - 1)(2x + 1)} = \frac{A}{2x - 1} + \frac{B}{2x + 1}\]

Here, \(A\) and \(B\) are constants to be determined later (but per the problem, we do not solve for them now).

This form is the appropriate setup for the partial fraction decomposition of the given expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex fraction into a sum of fractions with simpler denominators, typically linear or quadratic factors.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials or other simpler polynomials. Recognizing how to factor the denominator, such as difference of squares, is essential for setting up the correct form of partial fractions.

Form of Partial Fractions for Distinct Linear Factors

When the denominator factors into distinct linear terms, the partial fraction decomposition consists of separate fractions with unknown constants in the numerators over each linear factor. Setting up the correct form involves assigning constants to each linear factor without solving for them.

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.6. (4x + 1)/(4x² - 1)