Resolve \( \frac{1-x-x^{2}}{(1-x)\left(1-x^{3}\right)} \) into partial fractions.

To decompose the rational expression \(\frac{1 - x - x^{2}}{(1 - x)(1 - x^{3})}\) into partial fractions, follow these steps: 1. **Factor the Denominator:** Notice that \(1 - x^{3}\) can be factored as: \[ 1 - x^{3} = (1 - x)(1 + x + x^{2}) \] So, the denominator becomes: \[ (1 - x)(1 - x^{3}) = (1 - x)^{2}(1 + x + x^{2}) \] 2. **Set Up the Partial Fraction Decomposition:** The expression can be written as: \[ \frac{1 - x - x^{2}}{(1 - x)^{2}(1 + x + x^{2})} = \frac{A}{1 - x} + \frac{B}{(1 - x)^{2}} + \frac{Cx + D}{1 + x + x^{2}} \] 3. **Multiply Through by the Denominator:** Multiply both sides by \((1 - x)^{2}(1 + x + x^{2})\) to eliminate the denominators: \[ 1 - x - x^{2} = A(1 - x)(1 + x + x^{2}) + B(1 + x + x^{2}) + (Cx + D)(1 - x)^{2} \] 4. **Expand and Simplify:** Expand each term: \[ A(1 - x)(1 + x + x^{2}) = A(1 - x^{3}) \] \[ B(1 + x + x^{2}) = B + Bx + Bx^{2} \] \[ (Cx + D)(1 - x)^{2} = Cx - 2Cx^{2} + Cx^{3} + D - 2Dx + Dx^{2} \] Combine all terms: \[ 1 - x - x^{2} = (A + B + D) + (B + C - 2D)x + (B - 2C + D)x^{2} + (-A + C)x^{3} \] 5. **Set Up a System of Equations:** Equate the coefficients of like terms on both sides: \[ \begin{cases} -A + C = 0 \quad &\text{(from } x^{3}\text{ terms)} \\ B - 2C + D = -1 \quad &\text{(from } x^{2}\text{ terms)} \\ B + C - 2D = -1 \quad &\text{(from } x\text{ terms)} \\ A + B + D = 1 \quad &\text{(constant term)} \end{cases} \] 6. **Solve the System:** From the first equation: \[ C = A \] Substitute \(C = A\) into the remaining equations: \[ \begin{cases} B - 2A + D = -1 \\ B + A - 2D = -1 \\ A + B + D = 1 \end{cases} \] Solving these, we find: \[ A = \frac{2}{3}, \quad B = -\frac{1}{3}, \quad C = \frac{2}{3}, \quad D = \frac{2}{3} \] 7. **Write the Partial Fractions:** Substituting the values of \(A\), \(B\), \(C\), and \(D\) back into the partial fractions: \[ \frac{2}{3(1 - x)} - \frac{1}{3(1 - x)^{2}} + \frac{2x + 2}{3(1 + x + x^{2})} \] **Final Partial Fraction Decomposition:** \[ \boxed{\,\frac{2}{3(1 - x)}\; -\; \frac{1}{3\, (1 - x)^2}\; +\; \frac{2x + 2}{3(1 + x + x^{2})}\,} \]