Write the partial fraction decomposition for the expression....

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Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function, which is a ratio of two polynomials, as a sum of simpler rational expressions. This technique is particularly useful for integration and solving equations where dealing with a sum of simpler fractions is more manageable than working with a complex rational expression.

Repeated Linear Factors

When the denominator of a rational function consists of repeated linear factors, each factor raised to a power requires its own term in the decomposition. For example, if the denominator includes (x + a)^n, then the decomposition will include terms with denominators (x + a), (x + a)^2, ..., (x + a)^n. This accommodates the full structure of the repeated factors.

Equating Coefficients

The method of equating coefficients is used to solve for the unknown constants in a partial fraction decomposition. Once the rational expression is rewritten with undetermined constants in the numerators, both sides of the equation are manipulated into a common denominator, and the coefficients of corresponding powers of the variable are equated to form a system of linear equations. Solving this system yields the constants necessary for the decomposition.

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