1. **State the problem:** Find the roots of the cubic function $$f(x) = x^3 - 2x^2 + 4x - 8$$ and express it in intercept form. 2. **Number of roots:** A cubic polynomial has exactly 3 roots (real or complex) according to the Fundamental Theorem of Algebra. 3. **Find rational roots using the Rational Root Theorem:** Possible rational roots are factors of the constant term 8 divided by factors of the leading coefficient 1, i.e., $$\pm1, \pm2, \pm4, \pm8$$. 4. **Test rational roots by substitution:** - Test $$x=2$$: $$f(2) = 2^3 - 2(2)^2 + 4(2) - 8 = 8 - 8 + 8 - 8 = 0$$ So, $$x=2$$ is a root. 5. **Divide the polynomial by $$x-2$$ to find the quadratic factor:** Using synthetic division: $$\begin{array}{r|rrrr} 2 & 1 & -2 & 4 & -8 \\ & & 2 & 0 & 8 \\ \hline & 1 & 0 & 4 & 0 \\ \end{array}$$ The quotient is $$x^2 + 0x + 4 = x^2 + 4$$. 6. **Find roots of the quadratic factor:** Solve $$x^2 + 4 = 0$$ $$x^2 = -4$$ $$x = \pm \sqrt{-4} = \pm 2i$$ These are complex roots. 7. **Summary of roots:** - Rational root: $$x=2$$ - Complex roots: $$x=2i$$ and $$x=-2i$$ 8. **Write the intercept form:** $$f(x) = (x - 2)(x - 2i)(x + 2i)$$ Since $$(x - 2i)(x + 2i) = x^2 + 4$$, $$f(x) = (x - 2)(x^2 + 4)$$ **Final answers:** - Rational Roots: $$x=2$$ - Complex or Irrational Roots: $$x=\pm 2i$$ - Intercept Form: $$f(x) = (x - 2)(x^2 + 4)$$