Split Using Partial Fraction Decomposition (x^4+x^2+1)/(x^2(x^2+6)^2)

Problem

(x4+x2+1)/(x2*(x2+6)2)

Solution

1. Set up the partial fraction decomposition form based on the factors in the denominator. The term x2 is a repeated linear factor (treated as x squared), and (x2+6)2 is a repeated irreducible quadratic factor.

(x4+x2+1)/(x2*(x2+6)2)=A/x+B/(x2)+(C*x+D)/(x2+6)+(E*x+F)/((x2+6)2)

2. Multiply both sides by the common denominator x2*(x2+6)2 to clear the fractions.

x4+x2+1=A*x*(x2+6)2+B*(x2+6)2+(C*x+D)*x2*(x2+6)+(E*x+F)*x2

3. Expand the terms on the right side of the equation.

x4+x2+1=A*x*(x4+12*x2+36)+B*(x4+12*x2+36)+(C*x+D)*(x4+6*x2)+E*x3+F*x2

x4+x2+1=A*x5+12*A*x3+36*A*x+B*x4+12*B*x2+36*B+C*x5+6*C*x3+D*x4+6*D*x2+E*x3+F*x2

4. Group the terms by the powers of x

x4+x2+1=(A+C)*x5+(B+D)*x4+(12*A+6*C+E)*x3+(12*B+6*D+F)*x2+36*A*x+36*B

5. Equate the coefficients from both sides to create a system of equations.

A+C=0

B+D=1

12*A+6*C+E=0

12*B+6*D+F=1

36*A=0

36*B=1

6. Solve the system of equations starting with the simplest terms.

A=0

B=1/36

0+C=0⇒C=0

1/36+D=1⇒D=35/36

12*(0)+6*(0)+E=0⇒E=0

12*(1/36)+6*(35/36)+F=1⇒1/3+35/6+F=1⇒37/6+F=1⇒F=−31/6

7. Substitute the solved constants back into the decomposition form.

(x4+x2+1)/(x2*(x2+6)2)=0/x+(1/36)/(x2)+(0*x+35/36)/(x2+6)+(0*x−31/6)/((x2+6)2)

Final Answer

(x4+x2+1)/(x2*(x2+6)2)=1/(36*x2)+35/(36*(x2+6))−31/(6*(x2+6)2)

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