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Chapter 6: Techniques of Integration

Section 6.4: The Algebra of Partial Fractions

Example 6.4.5

Obtain the partial-fraction decomposition of $\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49}$ .

Solution

Mathematical Solution

The denominator, $x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49$ , factors to $(x + 1) {(x^{2} + 5 x - 7)}^{2}$ , so the decomposition begins with the identity

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} = \frac{a}{x + 1} + \frac{b_{1} x + c_{1}}{x^{2} + 5 x - 7} + \frac{b_{2} x + c_{2}}{{(x^{2} + 5 x - 7)}^{2}}$

There are many methods for determining the constants $a, b_{k}, c_{k}, k = 1, 2$ . The method adopted here is one that is amenable to an eventual Maple implementation of the underlying strategy. Multiply through by the factored denominator, obtaining

$5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109$	$= a {(x^{2} + 5 x - 7)}^{2} + (b_{1} x + c_{1}) (x + 1) (x^{2} + 5 x - 7) + (b_{2} x + c_{2}) (x + 1)$
	$= (a + b_{1}) x^{4} + (10 a + 6 b_{1} + c_{1}) x^{3} + (11 a - 2 b_{1} + b_{2} + 6 c_{1}) x^{2} + (- 70 a - 7 b_{1} + b_{2} - 2 c_{1} + c_{2}) x + 49 a - 7 c_{1} + c_{2}$

For this to be an identity in $x$ , the coefficients of like powers of $x$ on each side of the equal sign must match. Hence, the five equations

$a + b_{1}$	$= 5$
$10 a + 6 b_{1} + c_{1}$	$= 37$
$11 a - 2 b_{1} + b_{2} + 6 c_{1}$	$= 15$
$- 70 a - 7 b_{1} + b_{2} - 2 c_{1} + c_{2}$	$= - 150$
$49 a - 7 c_{1} + c_{2}$	$= 109$

must hold. The solution of these equations is $\{a = 2, b_{1} = 3, b_{2} = 5, c_{1} = - 1, c_{2} = 4\}$ , and the partial-fraction decomposition of the given rational function is

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} = \frac{2}{x + 1} + \frac{3 x - 1}{x^{2} + 5 x - 7} + \frac{5 x + 4}{{(x^{2} + 5 x - 7)}^{2}}$

Maple Solution

Solution by Context Panel

•	Control-drag the rational function.

•	Context Panel: Conversions≻Partial Fractions≻ $x$

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49}$ $\to$ $\frac{2}{x + 1} + \frac{3 x - 1}{x^{2} + 5 x - 7} + \frac{5 x + 4}{{(x^{2} + 5 x - 7)}^{2}}$

When working interactively, this solution via the Context Panel is probably the most efficient. Of course, there is the command-based conversion to partial fractions, as illustrated in Table 6.4.5(a).

•	Assign the name $f$ to the given rational function.

$f ≔ \frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} &colon;$

•	Apply the convert command, with the option parfrac, and the independent variable $x$ .

$convert (f, parfrac, x)$

$\frac{2}{x + 1} + \frac{3 x - 1}{x^{2} + 5 x - 7} + \frac{5 x + 4}{{(x^{2} + 5 x - 7)}^{2}}$

Table 6.4.5(a) Use of the convert command to obtain a partial-fraction decomposition

The stepwise partial-fractions task template in Table 6.4.5(b) provides an interactive stepwise solution.

Solution by Task Template

Tools≻Tasks≻Browse: Algebra≻Partial Fractions≻Stepwise

Stepwise Partial Fraction Decomposition

Write rational function here

$f (x) =$

$↓$

Write the partial-fraction decomposition template in this box

≡ *

$↓$

To determine the constants, multiply both sides of the identity (*) by the denominator of the fraction on the left.

$↓$

$=$

$↓$

$=$

$↓$

↓

$↓$

$f (x) =$ =

Table 6.4.5(b) Task template for a stepwise interactive partial-fraction decomposition

Table 6.4.5(c) contains an interactive solution from first principles. A similar calculation in the form of a recorded demo can be found here.

Interactive solution from first principles

•	Control-drag the given rational function.

•	Context Panel: Assign to a Name≻ $f$

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49}$ $\overset{assign to a name}{\to}$ $f$

•	Enter the decomposition template.

•	Context Panel: Assign to a Name≻ $g$

$\frac{a}{x + 1} + \frac{x b_{1} + c_{1}}{x^{2} + 5 x - 7} + \frac{x b_{2} + c_{2}}{{(x^{2} + 5 x - 7)}^{2}}$ $\overset{assign to a name}{\to}$ $g$

•	Write $f - g$ and press the Enter key.

•	Context Panel: Simplify≻Simplify

•	Context Panel: Numerator

•	Context Panel: Collect≻ $x$

•	Context Panel: Coefficients≻ $x$

•	Context Panel: Solve≻Solve

•	Context Panel: Assign to a Name≻ $S$

$f - g$

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} - \frac{a}{x + 1} - \frac{x b_{1} + c_{1}}{x^{2} + 5 x - 7} - \frac{x b_{2} + c_{2}}{{(x^{2} + 5 x - 7)}^{2}}$

$\overset{simplify}{=}$

$- \frac{a x^{4} + x^{4} b_{1} + 10 a x^{3} - 5 x^{4} + 6 x^{3} b_{1} + x^{3} c_{1} + 11 a x^{2} - 37 x^{3} - 2 x^{2} b_{1} + x^{2} b_{2} + 6 x^{2} c_{1} - 70 a x - 15 x^{2} - 7 x b_{1} + x b_{2} - 2 x c_{1} + x c_{2} + 49 a + 150 x - 7 c_{1} + c_{2} - 109}{(x + 1) {(x^{2} + 5 x - 7)}^{2}}$

$\overset{numerator}{\to}$

$- a x^{4} - x^{4} b_{1} - 10 a x^{3} + 5 x^{4} - 6 x^{3} b_{1} - x^{3} c_{1} - 11 a x^{2} + 37 x^{3} + 2 x^{2} b_{1} - x^{2} b_{2} - 6 x^{2} c_{1} + 70 a x + 15 x^{2} + 7 x b_{1} - x b_{2} + 2 x c_{1} - x c_{2} - 49 a - 150 x + 7 c_{1} - c_{2} + 109$

$\overset{collect w.r.t. x}{=}$

$(- a - b_{1} + 5) x^{4} + (- 10 a - 6 b_{1} - c_{1} + 37) x^{3} + (- 11 a + 2 b_{1} - b_{2} - 6 c_{1} + 15) x^{2} + (70 a + 7 b_{1} - b_{2} + 2 c_{1} - c_{2} - 150) x - 49 a + 7 c_{1} - c_{2} + 109$

$\overset{coefficients in x}{\to}$

$- 49 a + 7 c_{1} - c_{2} + 109, - 11 a + 2 b_{1} - b_{2} - 6 c_{1} + 15, - a - b_{1} + 5, - 10 a - 6 b_{1} - c_{1} + 37, 70 a + 7 b_{1} - b_{2} + 2 c_{1} - c_{2} - 150$

$\overset{solve}{\to}$

$\{a = 2, b_{1} = 3, b_{2} = 5, c_{1} = - 1, c_{2} = 4\}$

$\overset{assign to a name}{\to}$

$S$

•	Expression palette: Evaluation template Evaluate the template $g$ with the parameters in $S$

$\binom{g}{} | \binom{}{S}$ = $\frac{2}{x + 1} + \frac{3 x - 1}{x^{2} + 5 x - 7} + \frac{5 x + 4}{{(x^{2} + 5 x - 7)}^{2}}$

Table 6.4.5(c) Interactive solution from first principles

Table 6.4.5(d) contains a coded solution from first principles.

Coded solution from first principles

•	Write the basic identity for the partial-fraction decomposition.

$q_{1} ≔ f = g$

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} = \frac{a}{x + 1} + \frac{x b_{1} + c_{1}}{x^{2} + 5 x - 7} + \frac{x b_{2} + c_{2}}{{(x^{2} + 5 x - 7)}^{2}}$

•	Multiply through by the denominator of the rational function.

$q_{2} ≔ simplify (q_{1} \cdot denom (f))$

$5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109 = a x^{4} + x^{4} b_{1} + 10 a x^{3} + 6 x^{3} b_{1} + x^{3} c_{1} + 11 a x^{2} - 2 x^{2} b_{1} + x^{2} b_{2} + 6 x^{2} c_{1} - 70 a x - 7 x b_{1} + x b_{2} - 2 x c_{1} + x c_{2} + 49 a - 7 c_{1} + c_{2}$

•	Form the equation determined by identifying the coefficients of powers of $x^{0}$ .

$q_{3} ≔ map (coeff, q_{2}, x, 0)$

$109 = 49 a - 7 c_{1} + c_{2}$

•	Form the equation determined by identifying the coefficients of powers of $x^{1}$ .

$q_{4} ≔ map (coeff, q_{2}, x, 1)$

$- 150 = - 70 a - 7 b_{1} + b_{2} - 2 c_{1} + c_{2}$

•	Form the equation determined by identifying the coefficients of powers of $x^{2}$ .

$q_{5} ≔ map (coeff, q_{2}, x, 2)$

$15 = 11 a - 2 b_{1} + b_{2} + 6 c_{1}$

•	Form the equation determined by identifying the coefficients of powers of $x^{3}$ .

$q_{6} ≔ map (coeff, q_{2}, x, 3)$

$37 = 10 a + 6 b_{1} + c_{1}$

•	Form the equation determined by identifying the coefficients of powers of $x^{4}$ .

$q_{7} ≔ map (coeff, q_{2}, x, 4)$

$5 = a + b_{1}$

•	Solve the four equations in $a, b_{k}, c_{k}, k = 1, 2$ .

$q_{8} ≔ solve (\{q_{3}, q_{4}, q_{5}, q_{6}, q_{7}\})$

$\{a = 2, b_{1} = 3, b_{2} = 5, c_{1} = - 1, c_{2} = 4\}$

•	Evaluate the basic partial-fractions identity with the values of the coefficients determined by the solve command.

$eval (q_{1}, q_{8})$

$\frac{5 x^{4} + 37 x^{3} + 15 x^{2} - 150 x + 109}{x^{5} + 11 x^{4} + 21 x^{3} - 59 x^{2} - 21 x + 49} = \frac{2}{x + 1} + \frac{3 x - 1}{x^{2} + 5 x - 7} + \frac{5 x + 4}{{(x^{2} + 5 x - 7)}^{2}}$

Table 6.4.5(d) Coded solution from first principles

Essentially, the partial-fraction decomposition seeks to determine the parameters $a, b_{k}, c_{k}, k = 1, 2$ , that make equation $q_{1}$ (in Table 6.4.5(d)) an identity in $x$ . This can be done immediately with the following modification of Maple's solve command.

$solve (identity (q_{1}, x), \{a, b_{1}, c_{1}, b_{2}, c_{2}\})$

$\{a = 2, b_{1} = 3, b_{2} = 5, c_{1} = - 1, c_{2} = 4\}$

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