Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.15. x / ((x⁴ - 16)²)

Accepted Answer

Recognize that the denominator is $$ (x^4 - 16)^2 $$, which is a repeated factor. First, factor $$ x^4 - 16  as a $$difference of squares: $$ x^4 - 16 = (x^2 - 4)(x^2 + 4) . $$Further factor $$ x^2 - 4  as $$another difference of squares: $$ x^2 - 4 = (x - 2)(x + 2) . So $$the full factorization of the denominator is $$ (x - 2)(x + 2)(x^2 + 4) $$, and the entire denominator is $$ [(x - 2)(x + 2)(x^2 + 4)]^2 . $$Since the denominator is squared, each factor appears with multiplicity 2. For partial fractions, include terms for each power from 1 up to 2 for each factor. Set up the partial fraction decomposition as the sum of fractions with denominators $$ (x - 2) $$ and $$ (x - 2)^2 $$, $$ (x + 2) $$ and $$ (x + 2)^2 $$, and $$ (x^2 + 4) $$ and $$ (x^2 + 4)^2 . $$For linear factors, use constants in the numerators; for the irreducible quadratic $$ x^2 + 4 $$, use linear expressions in the numerator. Write the decomposition as:

$$ \frac{x}{(x^4 - 16)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{x + 2} + \frac{D}{(x + 2)^2} + \frac{Ex + F}{x^2 + 4} + \frac{Gx + H}{(x^2 + 4)^2} $$

where $$ A, B, C, D, E, F, G, H $$ are constants to be determined.

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
15. x / ((x⁴ - 16)²)

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Key Concepts

Partial Fraction Decomposition

Factorization of Polynomials

Handling Repeated Factors in Partial Fractions