题目
题目

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 47 (9.10, 9.11 and 9.12)

单项选择题

Which of the following equations describes the partial fraction decomposition of a rational function of the form 𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) ?

选项
A.𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) = 𝐴 𝑥 − 1 + 𝐵 𝑥 + 𝐶 ( 𝑥 − 1 ) 2 + 𝐷 𝑥 − 2 + 𝐸 𝑥 2 + 4
B.𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) = 𝐴 𝑥 − 1 + 𝐵 𝑥 ( 𝑥 − 1 ) 2 + 𝐶 𝑥 − 2 + 𝐷 𝑥 + 𝐸 𝑥 2 + 4
C.𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) = 𝐴 𝑥 − 1 + 𝐵 ( 𝑥 − 1 ) 2 + 𝐶 𝑥 − 2 + 𝐷 𝑥 𝑥 2 + 4
D.𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) = 𝐴 𝑥 − 1 + 𝐵 ( 𝑥 − 1 ) 2 + 𝐶 𝑥 − 2 + 𝐷 𝑥 + 𝐸 𝑥 2 + 4
E.𝑥 2 + 2 𝑥 + 3 ( 𝑥 − 1 ) 2 ( 𝑥 − 2 ) ( 𝑥 2 + 4 ) = 𝐴 𝑥 − 1 + 𝐵 ( 𝑥 − 1 ) 2 + 𝐶 𝑥 − 2 + 𝐷 𝑥 2 + 4
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思路分析
Question restatement: Which of the following equations describes the partial fraction decomposition of a rational function of the form (x^2 + 2x + 3) / [(x - 1)^2 (x - 2) (x^2 + 4)]? Option 1: (A)/(x - 1) + (B)/(x - 1)^2 + (C)/(x - 2) + (D x + E)/(x^2 + 4) - Why this could be plausible: This structure matches the standard decomposition for a denominator with a repeated linear factor (x - 1)^2, another linear factor (x - 2), and an irreducible quadratic (x^2 + 4). In a typical decomposition, we expect terms for: A/(x - 1), B/(x - 1)^2, C/(x - 2), and (Dx + E)/(x^2 + 4). - Why it might be wrong: The original option text appears to present A x − 1 and (x − 1)^2 in ways that don’t clearly align with the standard fractional form. If the numerators and structure are not clearly matching A/(x - 1), B/(x - 1)^2, C/(x - 2), and (Dx + E)/(x^2 + 4), it would ......Login to view full explanation

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