The task involves determining the correct representation of a rational function using the method of undetermined coefficients. Let's analyze the problem step by step.

Given Rational Function:

The rational function is: $\frac{2x^5 - 3x^3 + 11}{(x+2)^3(x^2+3)}$

Breakdown:

The denominator consists of two distinct factors:

1. $(x+2)^3$, which is a repeated linear factor.
2. $(x^2+3)$, which is an irreducible quadratic factor.

Using the method of undetermined coefficients, the rational function can be decomposed as:

For the repeated linear factor $(x+2)^3$:

We include terms for all powers of the factor up to its multiplicity. This gives: $\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3}$

For the irreducible quadratic factor $(x^2+3)$:

We include a single term of the form: $\frac{Dx + E}{x^2 + 3}$ Here, the numerator is a linear polynomial because the denominator is a quadratic polynomial.

Full Decomposition:

Combining both factors, the rational function is decomposed as: $\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3} + \frac{Dx + E}{x^2 + 3}$

Correct Answer:

From the options provided, the correct choice matches the fourth option: $\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3} + \frac{Dx + E}{x^2 + 3}$

Let me know if you'd like a more detailed breakdown of the process or have any further questions!

Relative Questions:

1. What is the method of undetermined coefficients used for in general?
2. Why do repeated linear factors require multiple terms in the decomposition?
3. How do irreducible quadratic factors affect the decomposition?
4. Can the coefficients $A, B, C, D, E$ be determined explicitly here?
5. How does partial fraction decomposition help in integration problems?

Tip:

When dealing with irreducible quadratic factors in partial fraction decomposition, always use a linear numerator for that term.