Question

The rational function rac{4x-1}{(x+1)(2x-3)}" alt="\frac{4x-1}{(x+1)(2x-3)}" align="absmiddle" class="latex-formula"> can be expressed as the sum of two partial fractions: $\frac{A}{x+1}$ and $\frac{B}{2x-3}$
Find the value of A-B:
3
1
-3
-1

Thank you!

Answer 1

Answer:

-1

Step-by-step explanation:

(4x − 1) / ((x + 1) (2x − 3)) = A / (x + 1) + B / (2x − 3)

Combine the right hand side into one fraction by finding the common denominator.

(A (2x − 3) + B (x + 1)) / ((x + 1) (2x − 3))

(2Ax − 3A + Bx + B) / ((x + 1) (2x − 3))

((2A + B)x + B − 3A) / ((x + 1) (2x − 3))

Set the numerator of this fraction equal to the numerator of the original fraction.

4x − 1 = (2A + B)x + B − 3A

Match the coefficients.

4 = 2A + B

-1 = B − 3A

Solve the system of equations.  Subtracting the second equation from the first:

5 = 5A

A = 1

B = 2

A − B = -1