In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

검증된 단계별 안내

Step 1: Recognize that the given rational expression \( \frac{x}{(x - 3)(x + 2)} \) is a proper fraction because the degree of the numerator (1) is less than the degree of the denominator (2). This means we can proceed with partial fraction decomposition.

Step 2: Set up the partial fraction decomposition. Since the denominator \((x - 3)(x + 2)\) consists of two distinct linear factors, the decomposition will take the form: \( \frac{x}{(x - 3)(x + 2)} = \frac{A}{x - 3} + \frac{B}{x + 2} \), where \(A\) and \(B\) are constants to be determined.

Step 3: Multiply through by the common denominator \((x - 3)(x + 2)\) to eliminate the fractions. This gives: \( x = A(x + 2) + B(x - 3) \).

Step 4: Expand and simplify the right-hand side. Distribute \(A\) and \(B\) to get: \( x = A \cdot x + 2A + B \cdot x - 3B \). Combine like terms: \( x = (A + B)x + (2A - 3B) \).

Step 5: Equate coefficients of like terms from both sides of the equation. For the \(x\)-terms: \( A + B = 1 \). For the constant terms: \( 2A - 3B = 0 \). Solve this system of linear equations to find \(A\) and \(B\). Substitute these values back into the partial fraction decomposition.

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주요 개념

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

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of this question.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational expressions or simplifying complex algebraic fractions, allowing for easier manipulation and analysis of the expression.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into its constituent factors, which can be linear or irreducible quadratic expressions. This process is essential for identifying the structure of the denominator in a rational expression, as it helps determine the form of the partial fractions needed for decomposition.