Partial Fraction Decomposition

Partial fraction decomposition is the process of breaking down a rational function into simpler fractions. In this case, we want to resolve (2x-5)/ (x 3)(x 1)² into partial fractions.

Step 1: Determine the degree of the denominator

The denominator of the fraction is (x 3)(x 1)². The degree is the highest power of x in the denominator, which is 3.

Step 2: Write the partial fraction decomposition

The partial fraction decomposition will have the form:

(2x-5)/ (x 3)(x 1)² = A/(x-3) + B/(x-1) + C/(x-1)²

where A, B, and C are constants that we need to solve for.

Step 3: Solve for the constants

To solve for the constants, we need to find a common denominator and then equate the numerators. This will give us a system of equations that we can solve for A, B, and C.

Multiplying both sides by (x-3)(x-1)², we get:

2x-5 = A(x-1)²(x-3) + B(x-3)(x-1)² + C(x-3)(x-1)²

Setting x = 3, we get:

-4 = 4A

A = -1

Setting x = 1, we get:

-3 = 4C

C = -3/4

Substituting these values back into the equation and simplifying, we get:

(2x-5)/ (x 3)(x 1)² = -1/(x-3) + B/(x-1) - 3/4/(x-1)²

To solve for B, we need to find a common denominator again and equate the numerators:

2x-5 = -1(x-1)(x-1) + B(x-3)(x-1)² - 3/4(x-3)(x-1)²

Setting x = 1, we get:

-3/4 = 4B

B = -3/16

Therefore, the partial fraction decomposition is:

(2x-5)/ (x 3)(x 1)² = -1/(x-3) - 3/16/(x-1) + -3/4/(x-1)²

Final Answer:

The final answer is:

(2x-5)/ (x 3)(x 1)² = -1/(x-3) - 3/16/(x-1) -3/4/(x-1)²