To resolve the expression \(\frac{3x-2}{(x^2+4)^2(x-1)}\) into partial fractions, we will follow these steps:
### Step 1: Set Up the Partial Fraction Decomposition
We express the given fraction as a sum of simpler fractions. The form will be:
\[
\frac{3x-2}{(x^2+4)^2(x-1)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+4} + \frac{Dx+E}{(x^2+4)^2}
\]
where \(A\), \(B\), \(C\), \(D\), and \(E\) are constants that we need to determine.
### Step 2: Combine the Right Side
To combine the right side into a single fraction, we take the least common multiple (LCM) of the denominators:
\[
\frac{A(x^2+4)^2 + (Bx+C)(x-1)(x^2+4) + (Dx+E)(x-1)}{(x^2+4)^2(x-1)}
\]
This must equal the left side:
\[
3x - 2 = A(x^2+4)^2 + (Bx+C)(x-1)(x^2+4) + (Dx+E)(x-1)
\]
### Step 3: Expand the Right Side
Now we expand the right side:
1. Expand \(A(x^2+4)^2\):
\[
A(x^4 + 8x^2 + 16)
\]
2. Expand \((Bx+C)(x-1)(x^2+4)\):
\[
(Bx+C)(x^3 + 4x - x^2 - 4) = (Bx+C)(x^3 - x^2 + 4x - 4)
\]
Expand this further.
3. Expand \((Dx+E)(x-1)\):
\[
Dx^2 + Ex - Dx - E = Dx^2 + (E-D)x - E
\]
### Step 4: Collect Like Terms
Combine all the terms from the expansions:
\[
3x - 2 = (A + B)x^4 + (8A - B + D)x^3 + (16A + 4B - 4C - D)x^2 + (8A + 4C + E - D)x + (16A - 4C - E)
\]
### Step 5: Set Up the System of Equations
Now, we equate the coefficients of \(x^4\), \(x^3\), \(x^2\), \(x\), and the constant term from both sides:
1. Coefficient of \(x^4\): \(A + B = 0\)
2. Coefficient of \(x^3\): \(8A - B + D = 0\)
3. Coefficient of \(x^2\): \(16A + 4B - 4C - D = 0\)
4. Coefficient of \(x\): \(8A + 4C + E - D = 3\)
5. Constant term: \(16A - 4C - E = -2\)
### Step 6: Solve the System of Equations
From \(A + B = 0\), we can express \(B = -A\).
Substituting \(B = -A\) into the other equations gives us a system of equations in terms of \(A\), \(C\), \(D\), and \(E\).
1. From \(8A - (-A) + D = 0\) → \(9A + D = 0\) → \(D = -9A\)
2. Substitute \(B\) and \(D\) into the other equations and solve for \(C\) and \(E\).
### Step 7: Substitute Back to Find Constants
After solving the equations, we will find the values of \(A\), \(B\), \(C\), \(D\), and \(E\).
### Final Step: Write the Partial Fraction Decomposition
Substituting the values of the constants back into the partial fraction decomposition gives us the final result.
