Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{x^3 + x^2}{(x^2 + 4)^2}$

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1. Identify the rational expression given: \(\frac{x^3 + x^2}{(x^2 + 4)^2}\).
2. Recognize that the denominator is a repeated irreducible quadratic factor: \((x^2 + 4)^2\).
3. For a repeated irreducible quadratic factor \((ax^2 + bx + c)^n\), the partial fraction decomposition includes terms of the form \(\frac{Bx + C}{ax^2 + bx + c}\), \(\frac{Dx + E}{(ax^2 + bx + c)^2}\), and so on, up to the power \(n\).
4. Write the general form of the partial fraction decomposition for this problem as: \(\frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2}\), where \(B\), \(C\), \(D\), and \(E\) are constants to be determined.
5. Note that since the numerator degree is less than the denominator degree, no polynomial division is necessary before setting up the partial fractions.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex rational expression into a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.

Repeated Quadratic Factors

When the denominator contains a repeated irreducible quadratic factor, such as (x² + 4)², the decomposition includes terms with the quadratic factor raised to each power up to its multiplicity. For example, terms with denominators (x² + 4) and (x² + 4)² appear, each with linear numerators.

Form of Numerators in Partial Fractions

For linear factors, numerators are constants; for irreducible quadratic factors, numerators are linear expressions (ax + b). This ensures the decomposition can represent any polynomial numerator of lower degree than the denominator factor, allowing for a complete and accurate breakdown.