Cubic Roots D95Aea

1. **State the problem:** Find all roots of the cubic function $$f(x) = 2x^3 - 2x^2 - 18x + 18$$. 2. **Formula and approach:** Roots of a polynomial are values of $x$ such that $f(x) = 0$. We solve: $$2x^3 - 2x^2 - 18x + 18 = 0$$ 3. **Step 1: Factor out the greatest common factor (GCF):** $$2(x^3 - x^2 - 9x + 9) = 0$$ Divide both sides by 2 (which does not affect roots): $$x^3 - x^2 - 9x + 9 = 0$$ 4. **Step 2: Use Rational Root Theorem to test possible roots:** Possible rational roots are factors of constant term 9 divided by factors of leading coefficient 1, i.e., $\pm1, \pm3, \pm9$. 5. **Step 3: Test $x=1$:** $$1^3 - 1^2 - 9(1) + 9 = 1 - 1 - 9 + 9 = 0$$ So, $x=1$ is a root. 6. **Step 4: Perform polynomial division or synthetic division to factor out $(x-1)$:** Divide $x^3 - x^2 - 9x + 9$ by $(x-1)$: Using synthetic division: \begin{align*} 1 &| 1 \quad -1 \quad -9 \quad 9 \\ &\quad 1 \quad 0 \quad -9 \\ \hline & 1 \quad 0 \quad -9 \quad 0 \end{align*} The quotient is $x^2 + 0x - 9 = x^2 - 9$. 7. **Step 5: Factor the quadratic:** $$x^2 - 9 = (x - 3)(x + 3)$$ 8. **Step 6: Write all roots:** The roots are $x = 1, 3, -3$. **Final answer:** $$\boxed{1, 3, -3}$$