Split Using Partial Fraction Decomposition (3x^2-7x-2)/(x^3-x)

Problem

(3*x2−7*x−2)/(x3−x)

Solution

1. Factor the denominator completely by first pulling out the greatest common factor x and then factoring the resulting difference of squares.

x3−x=x*(x2−1)

x3−x=x*(x−1)*(x+1)

2. Set up the partial fraction decomposition using three distinct linear factors, assigning a constant variable to each numerator.

(3*x2−7*x−2)/(x*(x−1)*(x+1))=A/x+B/(x−1)+C/(x+1)

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

3*x2−7*x−2=A*(x−1)*(x+1)+B*x*(x+1)+C*x*(x−1)

4. Solve for A by substituting x=0 into the equation.

3*(0)2−7*(0)−2=A*(0−1)*(0+1)+B(0)*(0+1)+C(0)*(0−1)

−2=−A

A=2

5. Solve for B by substituting x=1 into the equation.

3*(1)2−7*(1)−2=A*(1−1)*(1+1)+B(1)*(1+1)+C(1)*(1−1)

−6=2*B

B=−3

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

3*(−1)2−7*(−1)−2=A*(−1−1)*(−1+1)+B*(−1)*(−1+1)+C*(−1)*(−1−1)

3+7−2=2*C

8=2*C

C=4

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

Final Answer

(3*x2−7*x−2)/(x3−x)=2/x−3/(x−1)+4/(x+1)

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