x^3/(5x^3-1)
resolve into a partial fraction
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http://www.wolframalpha.com/input/?i=partial+fractions+x%5E3%2F(5x%5E3-1)
This one is kinda nasty, since 5x^3-1 does factor, but not with integer coefficients. It's a difference of two cubes, so it factors as
(∛5 x - 1)(∛25 x^2 + ∛5 x + 1))
To resolve the fraction x^3/(5x^3-1) into partial fractions, follow these steps:
Step 1: Factorize the denominator
The first step is to factorize the denominator, 5x^3-1. Notice that it is a difference of cubes, so we can use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2) to factorize it as:
5x^3 - 1 = (x - 1)(5x^2 + 5x + 1)
Step 2: Write the partial fraction form
The partial fraction form will have a numerator and factors in the denominator. In this case, given the factorization from step 1, the partial fraction form will be:
x^3/(5x^3-1) = A/(x - 1) + (Bx + C)/(5x^2 + 5x + 1)
Step 3: Solve for the unknown coefficients
To solve for the unknown coefficients A, B, and C, we need to express the numerator x^3 in terms of the partial fraction form. We can do this by multiplying both sides of the equation by the denominator (5x^3-1), giving:
x^3 = A(5x^2 + 5x + 1) + (Bx + C)(x - 1)
Step 4: Simplify and equate coefficients
Next, simplify the right-hand side of the equation and equate like terms. This will give us a system of equations to solve for the unknown coefficients A, B, and C.
For the x^3 term:
1 = 5A
For the x^2 term:
0 = 5A + B
For the x term:
0 = 5A + B - C
For the constant term (1):
0 = A - C
Step 5: Solve the system of equations
Solve the system of equations to find the values of A, B, and C. Solving the equations will give the following results:
From the first equation: A = 1/5
From the second equation: B = -5A = -1
From the third equation: C = 5A + B = 1
Step 6: Substitute the coefficients back into the partial fraction form
Now that we have found the values of A, B, and C, substitute these values back into the partial fraction form:
x^3/(5x^3 - 1) = 1/5/(x - 1) - (x + 1)/(5x^2 + 5x + 1)
So, the partial fraction decomposition of x^3/(5x^3 - 1) is 1/5(x - 1) - (x + 1)/(5x^2 + 5x + 1).