1. **Stating the problem:** We are given the cubic polynomial $x^3 - 2x^2 + 2kx + 8$ and need to analyze or solve it depending on the context (e.g., find roots, factor, or determine $k$ for certain conditions). 2. **Formula and rules:** For cubic polynomials of the form $ax^3 + bx^2 + cx + d$, the roots can be found using factoring, synthetic division, or the cubic formula. Important rules include the Rational Root Theorem and the factor theorem. 3. **Intermediate work:** To find roots or factor, we can try possible rational roots using factors of the constant term 8: $\pm1, \pm2, \pm4, \pm8$. 4. **Testing $x= -2$:** Substitute $x = -2$: $$(-2)^3 - 2(-2)^2 + 2k(-2) + 8 = -8 - 8 - 4k + 8 = -8 - 4k$$ For $x = -2$ to be a root, set equal to zero: $$-8 - 4k = 0 \implies 4k = -8 \implies k = -2$$ 5. **With $k = -2$, the polynomial becomes:** $$x^3 - 2x^2 - 4x + 8$$ 6. **Factor by synthetic division using root $x = -2$:** Divide $x^3 - 2x^2 - 4x + 8$ by $(x + 2)$: Coefficients: 1, -2, -4, 8 Bring down 1, multiply by -2: -2, add to -2: -4 Multiply -4 by -2: 8, add to -4: 4 Multiply 4 by -2: -8, add to 8: 0 (remainder) Quotient: $x^2 - 4x + 4$ 7. **Factor quotient:** $$x^2 - 4x + 4 = (x - 2)^2$$ 8. **Final factorization:** $$x^3 - 2x^2 - 4x + 8 = (x + 2)(x - 2)^2$$ 9. **Roots:** $$x = -2, 2 \text{ (double root)}$$ **Answer:** For $k = -2$, the polynomial factors as $(x + 2)(x - 2)^2$ with roots $-2$ and $2$ (double root).