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.9. 4/(x⁵ - 5x³ + 4x)

Accepted Answer

First, factor the denominator completely. The denominator is $$x^{5}$$ - $$5x^{3} + 4x$. Start by factoring out the common factor x$:  
x$$^{5} - 5x^{3} + 4x = x(x^{4} - 5x^{2} + 4)$$ Next, factor the quartic polynomial x$$^{4}$$ - 5x$$^{2}$$ + 4. $$Treat $$x^{2} as a $$variable, say $$y = x$$^{2}$$, so the expression becomes y$$^{2}$$ - 5y + 4. $$Factor this quadratic:  
$$y^{2} - 5y + 4 = (y - 4)(y - 1)$$  
Substitute back $$y = x$$^{2}$$ to get:  
 (x$$^{2} - 4)(x^{2} - 1) $$ Further factor the difference of squares:  
x$$^{2} - 4 = (x - 2)(x + 2)$$  
x$$^{2} - 1 = (x - 1)(x + 1)$$  
So the full factorization of the denominator is:  
x(x - 2)(x + 2)(x - 1)(x + 1$$)$$ Since all factors are linear and distinct, the partial fraction decomposition will have terms of the form:  
$$\frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} + \frac{D}{x - 1} + \frac{E}{x + 1}$$  
where A$, B$, C$, D$, and E$ are constants to be determined. Set up the equation:  
$$\frac{4}{x^{5} - 5x^{3} + 4x} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} + \frac{D}{x - 1} + \frac{E}{x + 1}$$  
This is the appropriate form of the partial fraction decomposition without solving for the constants.

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.
9. 4/(x⁵ - 5x³ + 4x)

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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Form of Partial Fractions for Repeated and Higher-Degree Factors