Question

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

Accepted Answer

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.

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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Key Concepts

Partial Fraction Decomposition

Handling Repeated Irreducible Quadratic Factors

Equating Coefficients to Solve for Unknowns