Chapter 6: Techniques of Integration
Section 6.4: The Algebra of Partial Fractions
Obtain the partial-fraction decomposition of 5⁢x4+37⁢x3+15⁢x2−150⁢x+109x5+11⁢x4+21⁢x3−59⁢x2−21⁢x+49.
The denominator, x5+11⁢x4+21⁢x3−59⁢x2−21 x+49, factors to x+1⁢x2+5⁢x−72, so the decomposition begins with the identity
5⁢x4+37⁢x3+15⁢x2−150⁢x+109x5+11⁢x4+21⁢x3−59⁢x2−21 x+49 =ax+1+b1 x+c1x2+5 x−7+b2 x+c2x2+5 x−72
There are many methods for determining the constants a, bk,ck,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
=a x2+5⁢x−72+ b1 x+c1 x+1 x2+5 x−7+b2 x+c2 x+1
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
must hold. The solution of these equations is a=2,b1=3,b2=5,c1=−1,c2=4, and the partial-fraction decomposition of the given rational function is
Solution by Context Panel
Control-drag the rational function.
Context Panel: Conversions≻Partial Fractions≻x
5⁢x4+37⁢x3+15⁢x2−150⁢x+109x5+11⁢x4+21⁢x3−59⁢x2−21 x+49 → 2x+1+3⁢x−1x2+5⁢x−7+5⁢x+4x2+5⁢x−72
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.
Apply the convert command, with the option parfrac, and the independent variable x.
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
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.
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
5⁢x4+37⁢x3+15⁢x2−150⁢x+109x5+11⁢x4+21⁢x3−59⁢x2−21 x+49→assign to a namef
Enter the decomposition template.
Context Panel: Assign to a Name≻g
ax+1+x⁢b1+c1x2+5⁢x−7+x⁢b2+c2x2+5⁢x−72→assign to a nameg
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
= collect w.r.t. x
→coefficients in x
→assign to a name
Expression palette: Evaluation template
Evaluate the template g with the parameters in S
gx=a|f(x)S = 2x+1+3⁢x−1x2+5⁢x−7+5⁢x+4x2+5⁢x−72
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.
Multiply through by the denominator of the rational function.
Form the equation determined by identifying the coefficients of powers of x0.
Form the equation determined by identifying the coefficients of powers of x1.
Form the equation determined by identifying the coefficients of powers of x2.
Form the equation determined by identifying the coefficients of powers of x3.
Form the equation determined by identifying the coefficients of powers of x4.
Solve the four equations in a,bk,ck,k=1,2.
Evaluate the basic partial-fractions identity with the values of the coefficients determined by the solve command.
Table 6.4.5(d) Coded solution from first principles
Essentially, the partial-fraction decomposition seeks to determine the parameters a,bk,ck,k=1,2, that make equation q1 (in Table 6.4.5(d)) an identity in x. This can be done immediately with the following modification of Maple's solve command.
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