To find the terms that appear in the partial fraction decomposition of the rational function

$F(s) = \frac{s^3 - 3}{(s^2 + 7)^2(s + 12)^2},$

we analyze its structure.

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Step 1: General Form of Partial Fraction Decomposition

For a denominator of the form $(s^2 + 7)^2(s + 12)^2$, the partial fraction decomposition includes:

1. Simple Roots and Linear Factors:

   • For $(s + 12)$, you include terms of the form: $\frac{G}{s + 12} + \frac{H}{(s + 12)^2}.$

2. Quadratic Irreducible Factors:

   • For $(s^2 + 7)$, since it's irreducible over the reals, you include terms of the form: $\frac{Bs + C}{s^2 + 7} + \frac{Es + F}{(s^2 + 7)^2}.$

3. Polynomial Numerators:

   • Each irreducible factor raised to a power $n$ contributes terms for all powers $1, 2, ..., n$.

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Step 2: Analyzing the Terms

The general partial fraction decomposition of $F(s)$ takes the form:

$F(s) = \frac{A}{s^2 + 7} + \frac{Bs + C}{(s^2 + 7)^2} + \frac{D}{s + 12} + \frac{E}{(s + 12)^2}.$

Adding potential generalizations, terms like $Ks + L$ may appear for certain decompositions.

Verifying the Given Terms

• A. $\frac{Ks + L}{(s + 12)^2}$:

  • A term of this form is valid if the numerator is a polynomial of degree less than the degree of the denominator. However, this is not required for this specific decomposition since only constants are needed.

• B. $\frac{J}{(s + 12)^2}$:

  • This term is valid because the denominator $(s + 12)^2$ corresponds to a partial fraction component.

• C. $\frac{D}{(s^2 + 7)^2}$:

  • This form should have a linear polynomial numerator (e.g., $Es + F$) for quadratic factors, so it does not directly apply.

• D. $\frac{G}{s + 12}$:

  • Valid because the factor $s + 12$ is simple in the decomposition.

• E. $\frac{Es + F}{(s^2 + 7)^2}$:

  • Valid because $(s^2 + 7)^2$ allows a linear numerator.

• F. $\frac{A}{s^2 + 7}$:

  • Valid because the factor $s^2 + 7$ is irreducible.

• G. $\frac{Hs + I}{s + 12}$:

  • Invalid because this does not align with the partial fraction decomposition rules for linear factors.

• H. $\frac{Bs + C}{s^2 + 7}$:

  • Valid because $(s^2 + 7)$ requires a linear numerator.

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Final Answer:

The terms that occur in the partial fraction decomposition are:

• B. $\frac{J}{(s + 12)^2}$
• D. $\frac{G}{s + 12}$
• E. $\frac{Es + F}{(s^2 + 7)^2}$
• F. $\frac{A}{s^2 + 7}$
• H. $\frac{Bs + C}{s^2 + 7}$

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Let me know if you'd like to see a detailed breakdown of any term!

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Questions for Further Understanding:

1. Why is a linear numerator used for quadratic irreducible factors in partial fraction decomposition?
2. What conditions dictate the powers of denominators in partial fraction terms?
3. How does the structure of the numerator influence the decomposition?
4. Can you identify cases where partial fraction decomposition is not applicable?
5. How would the decomposition change if $(s^2 + 7)^2$ were replaced by $(s^3 + 7)$?

Tip: Always verify irreducibility of quadratic terms over the real numbers before setting up the partial fraction decomposition!