Split Using Partial Fraction Decomposition (11x-2)/(x^2-x-12)

Problem

(11*x−2)/(x2−x−12)

Solution

1. Factor the denominator of the expression.

x2−x−12=(x−4)*(x+3)

2. Set up the partial fraction decomposition with unknown constants A and B

(11*x−2)/((x−4)*(x+3))=A/(x−4)+B/(x+3)

3. Multiply both sides by the common denominator (x−4)*(x+3) to clear the fractions.

11*x−2=A*(x+3)+B*(x−4)

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

11*(4)−2=A*(4+3)+B*(4−4)

42=7*A

A=6

5. Solve for B by substituting x=−3 into the equation.

11*(−3)−2=A*(−3+3)+B*(−3−4)

−35=−7*B

B=5

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

(11*x−2)/((x−4)*(x+3))=6/(x−4)+5/(x+3)

Final Answer

(11*x−2)/(x2−x−12)=6/(x−4)+5/(x+3)

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