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

Problem

(x+1)/((x−2)2*(x+4))

Solution

1. Set up the partial fraction decomposition by identifying the factors in the denominator. Since (x−2)2 is a repeated linear factor and (x+4) is a distinct linear factor, the decomposition takes the form:

(x+1)/((x−2)2*(x+4))=A/(x−2)+B/((x−2)2)+C/(x+4)

2. Clear the fractions by multiplying both sides of the equation by the common denominator (x−2)2*(x+4)

x+1=A*(x−2)*(x+4)+B*(x+4)+C*(x−2)2

3. Solve for B by substituting x=2 into the equation:

2+1=A(0)+B*(2+4)+C(0)

3=6*B

B=1/2

4. Solve for C by substituting x=−4 into the equation:

−4+1=A(0)+B(0)+C*(−4−2)2

−3=36*C

C=−1/12

5. Solve for A by comparing the coefficients of the x2 terms on both sides of the equation:

0=A+C

0=A−1/12

A=1/12

6. Substitute the values of A B and C back into the decomposition form:

(x+1)/((x−2)2*(x+4))=(1/12)/(x−2)+(1/2)/((x−2)2)−(1/12)/(x+4)

Final Answer

(x+1)/((x−2)2*(x+4))=1/(12*(x−2))+1/(2*(x−2)2)−1/(12*(x+4))

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