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
)
?

Question

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
)
?

BlackTom AI · Accepted Answer

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 not be the correct decomposition form.

Option 2: (A)/(x − 1) + (B)/(x + …) + (C)/(x − 1)² + (D)/(x^2 + 4)
- Why this could be plausible: It attempts to include a (x − 1)² term and a quadratic denominator, but the specific placement and powers do not align with the actual denominator (x − 2) and (x^2 + 4).
- Why it’s wrong: It misplaces the factors from the original denominator (x − 2) is present and needs a simple (x − 2) term, not a different linear factor or incorrect power distribution. It also uses a different numerator structure not matching the standard (Dx + E) over the irreducible quadratic.

Option 3: (A)/(x − 1) + (B)/(x − 1)² + (C)/(x − 2) + (D x)/(x^2 + 4) + (E)/(x^2 + 4)
- Why this could be plausible: It adopts the correct general decomposition shape: a term for (x − 1), a term for (x − 1)², a term for (x − 2), and a split over the irreducible quadratic into Dx/(x^2 + 4) and E/(x^2 + 4).
- Why it’s wrong: It omits the quotient (Dx + E)/(x^2 + 4) as a single combined fraction and separates it into Dx/(x^2 + 4) + E/(x^2 + 4). While algebraically equivalent, in the standard textbook form we typically present the irreducible quadratic part as (Dx + E)/(x^2 + 4). This option effectively expresses the same idea but with two separate fractions, which is not the conventional single-fraction decomposition.

Option 4: (A)/(x − 1) + (B)/(x − 1)² + (C)/(x − 2) + (D x)/(x^2 + 4) + (E)/(x^2 + 4)
- Why this could be plausible: Similar to Option 3, it includes all necessary denominator components: (x − 1) (with a simple and a repeated factor), (x − 2), and (x^2 + 4).
- Why it’s wrong: It splits the irreducible quadratic term into Dx/(x^2 + 4) and E/(x^2 + 4), which is not the standard consolidated form (Dx + E)/(x^2 + 4). Although algebraically legitimate, textbooks typically present a single irreducible-quadratic term rather than two separate fractions for the same denominator.

Option 5: (A)/(x − 1) + (B)(x − 1)² + (C)/(x − 2) + (D x)/(x^2 + 4) + (E)/(x^2 + 4)
- Why this could be plausible: It keeps the same factor structure and attempts to place constants with the appropriate denominators, including the irreducible quadratic part.
- Why it’s wrong: The second term shows B(x − 1)² in the numerator of a simple fraction form rather than B/(x − 1)², which misrepresents the coefficient structure for a repeated linear factor. This deviation from the standard A/(x−1) and B/(x−1)² form makes it inconsistent with the proper partial fraction template.

Overall reasoning: To correctly decompose a rational function with denominator (x − 1)² (x − 2) (x² + 4), the canonical form must include separate terms for A/(x − 1), B/(x − 1)², C/(x − 2), and a single irreducible quadratic term (Dx + E)/(x² + 4). Among the presented options, the one that clearly aligns with that canonical structure is the form that expresses the irreducible quadratic as (D x + E)/(x^2 + 4) while keeping A/(x − 1), B/(x − 1)², and C/(x − 2) as separate fractions. The other options either misplace factors, alter the standard numerators, or split the quadratic term into two separate fractions in a way that deviates from the conventional single-fraction irreducible-quadratic representation. Therefore, the correct decomposition form is the option that preserves A/(x − 1), B/(x − 1)², C/(x − 2), and (D x + E)/(x^2 + 4) in a single irreducible-quadratic term, i.e., the option that presents the last part as (D x + E)/(x^2 + 4).

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

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 ) ?

View Explanation

Log in for full answers

Similar Questions

In the partial fraction decomposition 2 𝑥 2 + 5 𝑥 + 9 ( 4 𝑥 + 1 ) ( 5 𝑥 2 + 1 ) = 𝐴 4 𝑥 + 1 + 𝐵 𝑥 + 𝐶 5 𝑥 2 + 1 , what are the values of 𝐴 , 𝐵 and 𝐶 ?

[math: 4x+1(x+4)2(x2+4)] \frac{4x+1}{(x+4)^2(x^2+4)} can be expressed in partial fractions as:

Calculate the integral $$\int_0^2 \dfrac{x+5}{x^2-2x-3}\, dx$$

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler