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

Problem

(4*x2+2*x−1)/(x2*(x+1))

Solution

1. Set up the partial fraction decomposition by identifying the factors in the denominator. Since x2 is a repeated linear factor and (x+1) is a distinct linear factor, the form is:

(4*x2+2*x−1)/(x2*(x+1))=A/x+B/(x2)+C/(x+1)

2. Clear the fractions by multiplying both sides of the equation by the common denominator x2*(x+1)

4*x2+2*x−1=A*x*(x+1)+B*(x+1)+C*x2

3. Solve for B by substituting x=0 into the equation:

4*(0)2+2*(0)−1=A(0)*(1)+B(1)+C*(0)2

−1=B

4. Solve for C by substituting x=−1 into the equation:

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

4−2−1=C

1=C

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

4*x2=A*x2+C*x2

4=A+C

4=A+1

3=A

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

(4*x2+2*x−1)/(x2*(x+1))=3/x+(−1)/(x2)+1/(x+1)

Final Answer

(4*x2+2*x−1)/(x2*(x+1))=3/x−1/(x2)+1/(x+1)

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