Resolve `(3x^(2)+x-2)/((x-2)^(2)(1-2x))` into partial fractions.

Answer

To resolve the expression \(\frac{3x^2 + x - 2}{(x-2)^2(1-2x)}\) into partial fractions, we follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the given fraction in terms of its partial fractions. The denominator consists of \((x-2)^2\) and \((1-2x)\), so we can write:

\[ \frac{3x^2 + x - 2}{(x-2)^2(1-2x)} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{1-2x} \]

### Step 2: Clear the Denominator
Next, we multiply both sides by the denominator \((x-2)^2(1-2x)\) to eliminate the fraction:

\[ 3x^2 + x - 2 = A(x-2)(1-2x) + B(1-2x) + C(x-2)^2 \]

### Step 3: Expand the Right Side
Now, we expand the right-hand side:

1. **Expand \(A(x-2)(1-2x)\)**:
\[ A(x-2)(1-2x) = A[(x-2) - 2x(x-2)] = A[x - 2 - 2x^2 + 4x] = A[-2x^2 + 5x - 2] \]

2. **Expand \(B(1-2x)\)**:
\[ B(1-2x) = B - 2Bx \]

3. **Expand \(C(x-2)^2\)**:
\[ C(x-2)^2 = C(x^2 - 4x + 4) = Cx^2 - 4Cx + 4C \]

Combining these expansions gives:

\[ 3x^2 + x - 2 = (-2A + C)x^2 + (5A - 2B - 4C)x + (-2A + B + 4C) \]

### Step 4: Compare Coefficients
Now, we compare coefficients from both sides:

1. For \(x^2\):
\[ -2A + C = 3 \quad \text{(1)} \]

2. For \(x\):
\[ 5A - 2B - 4C = 1 \quad \text{(2)} \]

3. For the constant term:
\[ -2A + B + 4C = -2 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
We now have a system of three equations:

1. From (1): \(C = 3 + 2A\)
2. Substitute \(C\) into (2):
\[ 5A - 2B - 4(3 + 2A) = 1 \implies 5A - 2B - 12 - 8A = 1 \implies -3A - 2B = 13 \quad \text{(4)} \]

3. Substitute \(C\) into (3):
\[ -2A + B + 4(3 + 2A) = -2 \implies -2A + B + 12 + 8A = -2 \implies 6A + B = -14 \quad \text{(5)} \]

Now we solve equations (4) and (5):

From (5):
\[ B = -14 - 6A \]

Substituting into (4):
\[ -3A - 2(-14 - 6A) = 13 \implies -3A + 28 + 12A = 13 \implies 9A = -15 \implies A = -\frac{5}{3} \]

Now substituting \(A\) back to find \(B\):
\[ B = -14 - 6\left(-\frac{5}{3}\right) = -14 + 10 = -4 \]

Now substituting \(A\) back to find \(C\):
\[ C = 3 + 2\left(-\frac{5}{3}\right) = 3 - \frac{10}{3} = -\frac{1}{3} \]

### Step 6: Write the Final Partial Fraction Decomposition
Now we have:
- \(A = -\frac{5}{3}\)
- \(B = -4\)
- \(C = -\frac{1}{3}\)

Thus, the partial fraction decomposition is:

\[ \frac{3x^2 + x - 2}{(x-2)^2(1-2x)} = \frac{-\frac{5}{3}}{x-2} + \frac{-4}{(x-2)^2} + \frac{-\frac{1}{3}}{1-2x} \]

### Final Answer:
\[ \frac{3x^2 + x - 2}{(x-2)^2(1-2x)} = -\frac{5}{3(x-2)} - \frac{4}{(x-2)^2} - \frac{1/3}{1-2x} \]