Write the form of the partial fraction decomposition of the rational expression.
2x-4/x^2-1
(2x-1)/(x^2-1) = a/(x+1) + b/(x-1)
2x-1 = ax-a + bx + b
2 = a+b
-1 = -a+b
-----------add
1 = 2 b
b = 1/2
a = 2-b = 3/2
so
(3/2) / (x+1) + (1/2)/(x-1)
To find the partial fraction decomposition of the rational expression 2x-4/x^2-1, follow these steps:
Step 1: Factor the denominator.
The denominator x^2-1 can be factored as (x+1)(x-1).
Step 2: Write the partial fraction decomposition.
The partial fraction decomposition is usually written in the form:
2x-4/x^2-1 = A/(x+1) + B/(x-1)
Step 3: Find the values of A and B.
To find A and B, we need to get a common denominator on the right side of the equation. Since the denominators are (x+1) and (x-1), the common denominator is (x+1)(x-1).
Now, multiply the entire equation by (x+1)(x-1):
2x-4 = A(x-1) + B(x+1)
Step 4: Solve for A and B.
Next, we expand the equation:
2x-4 = Ax - A + Bx + B
Comparing the coefficients of like terms, we have:
2x - 4 = (A + B) x + (-A + B)
By comparing the coefficients, we can set up a system of equations:
A + B = 2 (coefficient of x term)
-A + B = -4 (coefficient of constant term)
Solve this system of equations to find the values of A and B.