Chapter 6: Techniques of Integration
Section 6.4: The Algebra of Partial Fractions
Example 6.4.1
Execute the long division implied by the fraction x4−7 x2+5 x−8x2−2 x+3.
Solution
Mathematical Solution
Table 6.4.1(a) contains the long-division tableau that would be generated by a pencil-and-paper calculation. Note that an x3-term (with zero coefficient) is inserted into the dividend before the long division starts.
The x2 in the divisor is divided into x4 in the dividend, to produce the x2 in the quotient. This x2 in the quotient is multiplied by the divisor, and the resulting product is written beneath the dividend, and subtracted from the dividend to yield 2 x3−10 x. Immediately, bring down the 5 x from the dividend so that the newest line now becomes 2 x3−10 x2+5 x.
The x2 in the divisor is divided into the 2 x3 in the line 2 x3−10 x2+5 x and the resulting +2 x is added to the quotient. This 2 x is multiplied by the divisor, and the resulting product 2 x3−4 x2+6 x is written beneath the line 2 x3−10 x2+5 x and subtracted from that line to yield −6 x2−x. Immediately, bring down the −8 from the dividend so that the newest line becomes −6 x2−x−8.
The x2 in the divisor is divided into the −6 x2 in the line −6 x2−x−8 and the resulting −6 is appended to the quotient. This −6 is multiplied by the divisor, and the resulting product −6 x2+12 x−18 is written beneath the line −6 x2−x−8 and subtracted from that line to yield −13 x+10. Since the x2 in the divisor does not divide into −13 x, the long division terminates with −13 x+10 as the remainder.
Table 6.4.1(a) Long-division tableau
Maple Solution
The simplest way to effect the long division in Maple is to invoke a partial-fraction decomposition. As a step in the decomposition, Maple carries out the long division.
Control-drag the rational function.
Context Panel: Conversions≻Partial Fractions≻x
x4−7 x2+5 x−8x2−2 x+3 → x2+2x−6+−13x+10x2−2x+3
The quotient (x2+2 x−6) and remainder (−13 x+10) can be read from the decomposition. Since the zeros of the divisor x2−2 x+3 are the complex numbers 1 ±i 2, the divisor cannot be factored over the reals, and the fraction "remainder/divisor" cannot be further decomposed by the technique of partial fractions.
An alternate approach to determining the quotient and remainder, shown in Table 6.4.1(b), makes use of the quo command.
Context Panel: Numerator (Denominator)
Context Panel: Assign to a Name≻n (d)
x4−7 x2+5 x−8x2−2 x+3→numeratorx4−7x2+5x−8→assign to a namen
x4−7 x2+5 x−8x2−2 x+3→denominatorx2−2x+3→assign to a named
Apply the quo command.
Context Panel: Evaluate and Display Inline
quon,d,x,'r' = x2+2x−6
Type r.
r = −13x+10
Table 6.4.1(b) Use of the quo command to find the quotient and remainder
The letter to which the remainder is assigned does not have to be "r", but it needs to have the single quotes around it in the quo command. If this name is already assigned and is unquoted, then a syntax error will result. The quotes serve to unassign the name, allowing the quo command to attach a new remainder to the name.
A final alternative for obtaining the partial-fraction decomposition of a rational function is the convert command with the option parfrac.
convertx4−7 x2+5 x−8x2−2 x+3,parfrac,x = x2+2x−6+−13x+10x2−2x+3
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