Final answer:
To find the partial fraction decomposition, we factor the denominator as (2x)(x) and find the constants A and B using the equation 4x^2-3x+5 = A(x) + B(2x). Solving for the constants, we get A = -3 and B = 7. The partial fraction decomposition is (-3)/(2x) + 7/x.
Step-by-step explanation:
To find the partial fraction decomposition for the given expression, we first need to factor the denominator. The denominator is already a quadratic, so we can factor it as (2x)(x). Next, we need to find the constants A and B such that:
(4x^2-3x+5)/(2x^2-x) = A/(2x) + B/x
To find A and B, we can multiply both sides of the equation by the common denominator (2x)(x) to get:
4x^2-3x+5 = A(x) + B(2x)
Expanding and collecting like terms, we get:
4x^2 - 3x + 5 = (A + 2B)x + Ax
Equating the coefficients of x on both sides, we get:
4 = A + 2B and -3 = A
Solving this system of equations, we find A = -3 and B = 7. Therefore, the partial fraction decomposition of (4x^2-3x+5)/(2x^2-x) is:
(-3)/(2x) + 7/x