Exemplo 3 de Algebra and Trigonometry, 11.4 Partial Fractions

Decomposing P( x ) Q( x ) When Q(x) Contains a Nonrepeated Irreducible Quadratic Factor

Problema e solução em inglês, como no livro original.

Find a partial fraction decomposition of the given expression.

\frac{8 x^{2} + 12 x −20}{(x + 3) (x^{2} + x + 2)}

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

Solução
We have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant and the other numerator will be a linear expression. Thus,

$\frac{8 x^{2} + 12 x −20}{(x + 3) (x^{2} + x + 2)} = \frac{A}{(x + 3)} + \frac{B x + C}{(x^{2} + x + 2)}$
We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.

$\begin{array}{l} (x + 3) (x^{2} + x + 2) [\frac{8 x^{2} + 12 x - 20}{(x + 3) (x^{2} + x + 2)}] = [\frac{A}{(x + 3)} + \frac{B x + C}{(x^{2} + x + 2)}] (x + 3) (x^{2} + x + 2) \\ 8 x^{2} + 12 x - 20 = A (x^{2} + x + 2) + (B x + C) (x + 3) \end{array}$
Notice we could easily solve for $A$ by choosing a value for $x$ that will make the $B x + C$ term equal 0. Let $x = −3$ and substitute it into the equation.
$\begin{array}{l} 8 x^{2} + 12 x - 20 = A (x^{2} + x + 2) + (B x + C) (x + 3) \\ 8 {(- 3)}^{2} + 12 (- 3) - 20 = A ({(- 3)}^{2} + (- 3) + 2) + (B (- 3) + C) ((- 3) + 3) \\ 16 = 8 A \\ A = 2 \end{array}$
Now that we know the value of $A,$ substitute it back into the equation. Then expand the right side and collect like terms.

$\begin{array}{l} 8 x^{2} + 12 x −20 = 2 (x^{2} + x + 2) + (B x + C) (x + 3) \\ 8 x^{2} + 12 x −20 = 2 x^{2} + 2 x + 4 + B x^{2} + 3 B + C x + 3 C \\ 8 x^{2} + 12 x −20 = (2 + B) x^{2} + (2 + 3 B + C) x + (4 + 3 C) \end{array}$
Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.

$\begin{array}{l} 2 + B = 8 & (1) \\ 2 + 3 B + C = 12 & (2) \\ 4 + 3 C = −20 & (3) \end{array}$
Solve for $B$ using equation (1) and solve for $C$ using equation (3).

$\begin{array}{l} 2 + B = 8 & (1) \\ B = 6 \\ 4 + 3 C = −20 & (3) \\ 3 C = −24 \\ C = −8 \end{array}$
Thus, the partial fraction decomposition of the expression is

$\frac{8 x^{2} + 12 x −20}{(x + 3) (x^{2} + x + 2)} = \frac{2}{(x + 3)} + \frac{6 x −8}{(x^{2} + x + 2)}$

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

\frac{5 x^{2} −6 x + 7}{(x −1) (x^{2} + 1)}

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