Thanks you kindly. Express (2x^2-5x-5)/((2x^2+5)(4x-5)) in partial fractions.
To express the given rational expression (2x^2-5x-5)/((2x^2+5)(4x-5)) in partial fractions, follow these steps:
Step 1: Factorize the denominator
The denominator (2x^2+5)(4x-5) is already factorized, as it is the product of two distinct linear factors.
Step 2: Set up the partial fraction decomposition
Let's assume the partial fractions have the following form:
(2x^2-5x-5)/((2x^2+5)(4x-5)) = A/(2x^2+5) + B/(4x-5)
Step 3: Clear the fractions by multiplying through by the common denominator
(2x^2-5x-5) = A(4x-5) + B(2x^2+5)
Step 4: Expand the equation and collect like terms
2x^2 - 5x - 5 = (4A + 2B)x^2 + (-5A)x + (5B - 5)
Now, equate the coefficients of the like terms, which will lead to a system of equations.
Comparing the like terms on both sides of the equation, we get:
Coefficient of x^2: 2 = 4A + 2B
Coefficient of x: -5 = -5A
Constant term: -5 = 5B - 5
Step 5: Solve the system of equations
From the second equation, -5A = -5, we find A = 1.
Substituting A = 1 in the first equation, 2 = 4(1) + 2B gives 2 = 4 + 2B.
Simplifying, we get 2B = -2, which means B = -1.
Therefore, A = 1 and B = -1.
Step 6: Express the original expression in partial fractions
Now that we have obtained the values of A and B, we can write the original expression in partial fraction form:
(2x^2-5x-5)/((2x^2+5)(4x-5)) = 1/(2x^2+5) - 1/(4x-5)
And that's the partial fraction decomposition of the given rational expression!