Write the partial fraction decomposition of each rational expression. $\frac{x^3 + x^2 + 2}{(x^2 + 2)^2}$

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1. Identify the given rational expression: \(\frac{x^3 + x^2 + 2}{(x^2 + 2)^2}\).
2. Note that the denominator is a repeated irreducible quadratic factor, \((x^2 + 2)^2\).
3. Set up the partial fraction decomposition form for a repeated irreducible quadratic: \(\frac{x^3 + x^2 + 2}{(x^2 + 2)^2} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{(x^2 + 2)^2}\), where \(Ax + B\) and \(Cx + D\) are linear numerators because the denominator factors are quadratic and irreducible.
4. Multiply both sides of the equation by the common denominator \((x^2 + 2)^2\) to clear the fractions: \(x^3 + x^2 + 2 = (Ax + B)(x^2 + 2) + (Cx + D)\).
5. Expand the right side, collect like terms, and then equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations to solve for \(A\), \(B\), \(C\), and \(D\).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions with denominators that are factors of the original denominator. This technique is useful for integration and solving equations involving rational expressions.

Repeated Quadratic Factors

When the denominator contains repeated irreducible quadratic factors, each power of the factor must be included in the decomposition with numerators as linear expressions. This ensures the decomposition accounts for all degrees of the repeated factor.