Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)

검증된 단계별 안내

Recognize that the denominator \(x^{2} - 2x + 1\) can be factored. Factor it as a perfect square: \(x^{2} - 2x + 1 = (x - 1)^{2}\).

Set up the partial fraction decomposition for the rational expression \(\frac{2x + 2}{(x - 1)^{2}}\). Since the denominator is a repeated linear factor, express it as \(\frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\).

Multiply both sides of the equation by the denominator \((x - 1)^{2}\) to clear the fractions: \(2x + 2 = A(x - 1) + B\).

Expand the right side: \(2x + 2 = A x - A + B\). Then, group like terms: \(2x + 2 = A x + (B - A)\).

Equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations: For \(x\), \(2 = A\); for the constant term, \(2 = B - A\). Solve this system to find \(A\) and \(B\).

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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 simpler terms with denominators that are factors of the original denominator.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials or identifying perfect square trinomials. Recognizing the factorization of the denominator is essential for setting up the partial fractions correctly.

Handling Repeated Factors in Denominators

When the denominator has repeated factors, such as (x - a)², partial fractions must include terms for each power of the repeated factor. This ensures the decomposition accounts for all components of the denominator for accurate expansion.

Expand the quotients in Exercises 1–8 by partial fractions.(2x + 2) / (x² - 2x + 1)