Write the Form of the Partial Fraction Decomposition of the Rational

question 3

Multiple Choice

Write the form of the partial fraction decomposition of the rational expression.Do not solve for the constants. $\frac { x + 5 } { x \left( x ^ { 2 } + 6 \right) ^ { 2 } }$

• A) \(\frac { A } { x } + \frac { B x + C } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }\)
• B) \(\frac { A x + B } { x ^ { 2 } + 6 } + \frac { C x + D } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }\)
• C) \(\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 6 } + \frac { D x + E } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }\)
• D) \(\frac { A x + B } { x } + \frac { C x + D } { x ^ { 2 } + 6 } + \frac { E x + F } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }\)
• E) \(\frac { A } { x } + \frac { B } { x ^ { 2 } + 6 } + \frac { C } { \left( x ^ { 2 } + 6 \right) ^ { 2 } }\)

---

Definitions:

Free Energy

A thermodynamic quantity equivalent to the capacity of a system to do work, often used in biochemical and metabolic contexts.

Metabolic Energy

The energy that is produced and used by the cells during metabolic processes, necessary for maintaining the body's functions, such as muscle contractions and temperature regulation.

Potential Energy

The stored energy in an object or system due to its position or configuration.

Positron Emission Tomography

A type of medical imaging technique that uses radioactive substances to visualize and measure changes in metabolic processes and other physiological activities in the body.