Exemplo 4 de Algebra and Trigonometry, 11.4 Partial Fractions

Decomposing a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator

Problema e solução em inglês, como no livro original.

Decompose the given expression that has a repeated irreducible factor in the denominator.

\frac{x^{4} + x^{3} + x^{2} - x + 1}{x {(x^{2} + 1)}^{2}}

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
The factors of the denominator are $x, (x^{2} + 1),$ and ${(x^{2} + 1)}^{2} .$ Recall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear form $A x + B .$ So, let’s begin the decomposition.

$\frac{x^{4} + x^{3} + x^{2} - x + 1}{x {(x^{2} + 1)}^{2}} = \frac{A}{x} + \frac{B x + C}{(x^{2} + 1)} + \frac{D x + E}{{(x^{2} + 1)}^{2}}$

We eliminate the denominators by multiplying each term by $x {(x^{2} + 1)}^{2} .$ Thus,

$x^{4} + x^{3} + x^{2} - x + 1 = A {(x^{2} + 1)}^{2} + (B x + C) (x) (x^{2} + 1) + (D x + E) (x)$
Expand the right side.

$\begin{array}{l} x^{4} + x^{3} + x^{2} - x + 1 = A (x^{4} + 2 x^{2} + 1) + B x^{4} + B x^{2} + C x^{3} + C x + D x^{2} + E x \\ = A x^{4} + 2 A x^{2} + A + B x^{4} + B x^{2} + C x^{3} + C x + D x^{2} + E x \end{array}$
Now we will collect like terms.

$x^{4} + x^{3} + x^{2} - x + 1 = (A + B) x^{4} + (C) x^{3} + (2 A + B + D) x^{2} + (C + E) x + A$

Set up the system of equations matching corresponding coefficients on each side of the equal sign.

$\begin{array}{l} A + B = 1 \\ C = 1 \\ 2 A + B + D = 1 \\ C + E = −1 \\ A = 1 \end{array}$
We can use substitution from this point. Substitute $A = 1$ into the first equation.

$\begin{array}{l} 1 + B = 1 \\ B = 0 \end{array}$
Substitute $A = 1$ and $B = 0$ into the third equation.

$\begin{array}{l} 2 (1) + 0 + D = 1 \\ D = −1 \end{array}$
Substitute $C = 1$ into the fourth equation.

$\begin{array}{r} 1 + E = −1 \\ E = −2 \end{array}$
Now we have solved for all of the unknowns on the right side of the equal sign. We have $A = 1,$ $B = 0,$ $C = 1,$ $D = −1,$ and $E = −2.$ We can write the decomposition as follows:
$\frac{x^{4} + x^{3} + x^{2} - x + 1}{x {(x^{2} + 1)}^{2}} = \frac{1}{x} + \frac{1}{(x^{2} + 1)} - \frac{x + 2}{{(x^{2} + 1)}^{2}}$

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Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.

\frac{x^{3} −4 x^{2} + 9 x −5}{{(x^{2} −2 x + 3)}^{2}}

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