Partial fractions of `(x-2)/((1-x^2)(1-2x))="........"`

Answer

To find the partial fractions of the expression \(\frac{x-2}{(1-x^2)(1-2x)}\), we will follow these steps:

### Step 1: Factor the Denominator
The denominator can be factored as follows:
\[ 1 - x^2 = (1 - x)(1 + x) \]
Thus, we can rewrite the expression:
\[ \frac{x-2}{(1-x)(1+x)(1-2x)} \]

### Step 2: Set Up the Partial Fraction Decomposition
We can express the fraction as a sum of partial fractions:
\[ \frac{x-2}{(1-x)(1+x)(1-2x)} = \frac{A}{1-x} + \frac{B}{1+x} + \frac{C}{1-2x} \]
where \(A\), \(B\), and \(C\) are constants that we need to determine.

### Step 3: Clear the Denominator
Multiply both sides by the denominator \((1-x)(1+x)(1-2x)\) to eliminate the fractions:
\[ x - 2 = A(1+x)(1-2x) + B(1-x)(1-2x) + C(1-x)(1+x) \]

### Step 4: Expand the Right Side
Now, expand the right-hand side:
1. For \(A(1+x)(1-2x)\):
\[ A(1 - 2x + x - 2x^2) = A(1 - x - 2x^2) \]
2. For \(B(1-x)(1-2x)\):
\[ B(1 - 2x - x + 2x^2) = B(1 - 3x + 2x^2) \]
3. For \(C(1-x)(1+x)\):
\[ C(1 - x^2) \]

Combining these, we get:
\[ x - 2 = A(1 - x - 2x^2) + B(1 - 3x + 2x^2) + C(1 - x^2) \]

### Step 5: Collect Like Terms
Combine all terms on the right side:
\[ x - 2 = (A + B + C) + (-A - 3B)x + (-2A + 2B - C)x^2 \]

### Step 6: Set Up the System of Equations
Now, we equate the coefficients from both sides:
1. Constant term: \(A + B + C = -2\)
2. Coefficient of \(x\): \(-A - 3B = 1\)
3. Coefficient of \(x^2\): \(-2A + 2B - C = 0\)

### Step 7: Solve the System of Equations
We can solve this system of equations step by step.

1. From equation 1: \(C = -2 - A - B\)
2. Substitute \(C\) into equation 3:
\[ -2A + 2B - (-2 - A - B) = 0 \implies -2A + 2B + 2 + A + B = 0 \implies -A + 3B + 2 = 0 \implies A = 3B + 2 \]
3. Substitute \(A\) into equation 2:
\[ -(3B + 2) - 3B = 1 \implies -3B - 2 - 3B = 1 \implies -6B - 2 = 1 \implies -6B = 3 \implies B = -\frac{1}{2} \]
4. Substitute \(B\) back to find \(A\):
\[ A = 3(-\frac{1}{2}) + 2 = -\frac{3}{2} + 2 = \frac{1}{2} \]
5. Finally, substitute \(A\) and \(B\) to find \(C\):
\[ C = -2 - \frac{1}{2} + \frac{1}{2} = -2 \]

### Step 8: Write the Final Partial Fraction Decomposition
Now we have:
\[ A = \frac{1}{2}, \quad B = -\frac{1}{2}, \quad C = -2 \]
Thus, the partial fraction decomposition is:
\[ \frac{x-2}{(1-x)(1+x)(1-2x)} = \frac{1/2}{1-x} - \frac{1/2}{1+x} - \frac{2}{1-2x} \]

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