Find the Partial Fraction Decomposition for the Rational Expression $$\frac { 4 x ^ { 3 } - 22 x ^ { 2 } + 33 x - 6 } { ( x - 2 ) ^ { 3 } ( x + 2 ) }$$

question 342

Multiple Choice

Find the partial fraction decomposition for the rational expression.
- $$\frac { 4 x ^ { 3 } - 22 x ^ { 2 } + 33 x - 6 } { ( x - 2 ) ^ { 3 } ( x + 2 ) }$$

• A) \(\frac { 1 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { 1 } { ( x - 2 ) ^ { 3 } } + \frac { 3 } { x + 2 }\)
• B) \(\frac { 3 } { x - 2 } + \frac { - 1 } { ( x - 2 ) ^ { 2 } } + \frac { - 2 } { ( x - 2 ) ^ { 3 } } + \frac { 1 } { x + 2 }\)
• C) \(\frac { 1 } { x - 2 } + \frac { - 2 } { ( x - 2 ) ^ { 2 } } + \frac { 1 } { ( x - 2 ) ^ { 3 } } + \frac { 3 } { x + 2 }\)
• D) \(\frac { 1 } { x - 2 } + \frac { - 1 } { ( x - 2 ) ^ { 2 } } + \frac { 2 } { ( x - 2 ) ^ { 3 } } + \frac { - 1 } { x + 2 }\)

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Definitions:

Curiosity

A strong desire to know or learn something, driving exploration and investigation.

Independent Variable

In experimental research, the variable that is manipulated by the researcher to examine its effect on the dependent variable.

Experimental Study

A research method where the researcher manipulates one variable to determine if it causes a change in another variable, within a controlled environment.

Dependent Variable

The variable in an experiment that is expected to change as a result of changes made to the independent variable.