Math Problem Statement

Give the appropriate form of the partial fraction decomposition for the following function. StartFraction 2 x Over left parenthesis x minus 2 right parenthesis squared left parenthesis x squared plus 1 right parenthesis EndFraction Question content area bottom Part 1 Choose the correct answer below. A.StartFraction Upper A Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Bx plus Upper C Over x squared plus 1 EndFraction StartFraction Upper A Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Bx plus Upper C Over x squared plus 1 EndFraction B.StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Upper B Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Cx plus Upper D Over x squared plus 1 EndFraction StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Upper B Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Cx plus Upper D Over x squared plus 1 EndFraction C.StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Upper B Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Upper C Over x squared plus 1 EndFraction StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Upper B Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Upper C Over x squared plus 1 EndFraction D.StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Bx plus Upper C Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Dx plus Upper E Over x squared plus 1 EndFraction StartFraction Upper A Over x minus 2 EndFraction plus StartFraction Bx plus Upper C Over left parenthesis x minus 2 right parenthesis squared EndFraction plus StartFraction Dx plus Upper E Over x squared plus 1 EndFraction

Solution

The function you provided is:

2x(x−2)2(x2+1)\frac{2x}{(x - 2)^2(x^2 + 1)}

To find the appropriate form for the partial fraction decomposition, we follow these guidelines:

  1. For the (x−2)2(x - 2)^2 term: Since this is a repeated linear factor, we must account for both the first power (x−2)(x - 2) and the second power (x−2)2(x - 2)^2.
  2. For the (x2+1)(x^2 + 1) term: This is an irreducible quadratic factor, so the numerator should be linear in the form Cx+DCx + D.

Based on these considerations, the correct form of the partial fraction decomposition is:

Ax−2+B(x−2)2+Cx+Dx2+1\frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{Cx + D}{x^2 + 1}

This corresponds to Option B:

Ax−2+B(x−2)2+Cx+Dx2+1\frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{Cx + D}{x^2 + 1}

Do you want more details on how to solve this step-by-step?

Here are 5 related questions you can explore:

  1. How do you find the constants AA, BB, CC, and DD in partial fraction decomposition?
  2. What is the difference between linear and irreducible quadratic factors in partial fraction decomposition?
  3. How do repeated factors affect the form of partial fractions?
  4. Can partial fractions be applied to improper fractions, and how is it done?
  5. What role does the degree of the numerator and denominator play in determining the form of the decomposition?

Tip: Always start by factoring the denominator completely before setting up the partial fractions.

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Math Problem Analysis

Mathematical Concepts

Partial Fraction Decomposition
Rational Functions
Quadratic and Linear Factors

Formulas

Partial Fraction Decomposition: A / (x - r) + B / (x - r)^2 + (Cx + D) / (x^2 + 1)
Factorization: (x - 2)^2 and (x^2 + 1)

Theorems

Partial Fraction Theorem: Decomposing a rational function into simpler fractions with distinct and repeated factors
Factorization of Polynomials: Handling linear and irreducible quadratic factors

Suitable Grade Level

Grades 11-12, College level

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