Engineering Your Future
Engineering Your Future
9th Edition
ISBN: 9780190279264
Author: William C. Oakes, Les L. Leone
Publisher: Oxford University Press, USA
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Chapter 16, Problem 16.7EAA
To determine

Partial fraction decomposition of the given rational expression.

Expert Solution & Answer
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Answer to Problem 16.7EAA

The required partial fraction decomposition of the given rational expression is 4x2(x2)2=1x+1x21(x2)+1(x2)2.

Therefore, option (b) is the correct answer.

Explanation of Solution

Given information:

A rational expression: 4x2(x24x+4).

Calculation:

First of all, we factor the quadratic in the denominator:

x24x+4=x22x2x+4 (Splitting middle term)x24x+4=x(x2)2(x2) (Pulling GCF from each pair)x24x+4=(x2)(x2)x24x+4=(x2)2.

The given rational expression becomes: 4x2(x2)2.

Now we set up the partial fraction composition as shown below:

4x2(x2)2=Ax+Bx2+C(x2)+D(x2)2.

Upon multiplying by x2(x2)2 throughout this equation, we get:

4=Ax(x2)2+B(x2)2+Cx2(x2)+Dx2.

Now we need to find the values of A, B, C and D by substituting some values of x.

At x=04=A(0)(02)2+B(02)2+C(0)2(02)+D(0)24B=4B=1At x=24=A(2)(22)2+B(22)2+C(2)2(22)+D(2)24D=4D=1At x=14=A(1)(12)2+B(12)2+C(1)2(12)+D(1)24=A+BC+D4=A+1C+1AC=2At x=34=A(3)(32)2+B(32)2+C(3)2(32)+D(3)24=3A+B+9C+9D3A+1+9C+9(1)=43A+9C=6A+3C=2.

We have the values of B and D. In order to get the values of A and C, we solve the system of linear equations: AC=2, A+3C=2 . Upon solving this system, we get: A=1,C=1.

Upon substituting A=1,B=1,C=1,D=1 in our set up for partial fractions, we get:

4x2(x2)2=1x+1x2+1(x2)+1(x2)24x2(x2)2=1x+1x21(x2)+1(x2)2.

Conclusion:

The required partial fraction decomposition of the given rational expression is 4x2(x2)2=1x+1x21(x2)+1(x2)2.

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