Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-x⁴ - 8x² + 3x - 10)/((x + 2)(x² + 4)²)

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Identify the form of the denominator and set up the partial fraction decomposition accordingly. The denominator is \((x + 2)(x^2 + 4)^2\), which includes a linear factor \(x + 2\) and a repeated irreducible quadratic factor \((x^2 + 4)^2\).

Write the general form of the partial fraction decomposition: \[ \frac{-x^4 - 8x^2 + 3x - 10}{(x + 2)(x^2 + 4)^2} = \frac{A}{x + 2} + \frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2} \] Here, \(A, B, C, D, E\) are constants to be determined.

Multiply both sides of the equation by the denominator \((x + 2)(x^2 + 4)^2\) to clear the fractions, resulting in: \[ -x^4 - 8x^2 + 3x - 10 = A(x^2 + 4)^2 + (Bx + C)(x + 2)(x^2 + 4) + (Dx + E)(x + 2) \]

Expand the right-hand side by first expanding \((x^2 + 4)^2\), then distributing \((Bx + C)(x + 2)(x^2 + 4)\), and finally \((Dx + E)(x + 2)\). Combine like terms to express the right side as a polynomial in powers of \(x\).

Equate the coefficients of corresponding powers of \(x\) from both sides of the equation to form a system of linear equations. Solve this system for \(A, B, C, D, E\) to find the constants for the partial fraction decomposition.

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Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or simplifying expressions. It involves breaking down the denominator into linear and irreducible quadratic factors and assigning appropriate numerators to each.

Handling Repeated Irreducible Quadratic Factors

When the denominator contains repeated irreducible quadratic factors, such as (x^2 + 4)^2, the decomposition includes terms with increasing powers of that quadratic in the denominator. Each term has a linear numerator (Ax + B) to account for all possible numerators, ensuring the decomposition is complete.

Equating Coefficients to Solve for Unknowns

After setting up the partial fractions, multiply both sides by the common denominator to clear fractions. Then, expand and collect like terms to form a polynomial equation. Equate the coefficients of corresponding powers of x on both sides to create a system of equations, which can be solved to find the unknown constants in the numerators.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-x4 - 8x2 + 3x - 10)/((x + 2)(x2 + 4)2)