1. **Problem statement:** We have a cubic function $$f(x) = ax^3 + bx^2 + cx - 8$$ with zeros at $$x = -1$$ and $$x = 2$$, and it passes through the point $$(1, -2)$$. We need to: a) Find the remaining zero. b) State the domain and range. c) Sketch the function and fully label the graph. --- 2. **Using the zeros to factor the function:** Since $$x = -1$$ and $$x = 2$$ are zeros, $$f(x)$$ can be factored as: $$f(x) = a(x + 1)(x - 2)(x - r)$$ where $$r$$ is the remaining zero we want to find. 3. **Using the constant term:** The constant term in $$f(x)$$ is $$-8$$. Expanding the factors at $$x=0$$: $$f(0) = a(0 + 1)(0 - 2)(0 - r) = a(1)(-2)(-r) = 2ar$$ Given $$f(0) = -8$$, we have: $$2ar = -8$$ 4. **Using the point (1, -2):** Substitute $$x=1$$ and $$f(1) = -2$$: $$f(1) = a(1 + 1)(1 - 2)(1 - r) = a(2)(-1)(1 - r) = -2a(1 - r)$$ Set equal to $$-2$$: $$-2a(1 - r) = -2$$ Divide both sides by $$-2$$: $$\cancel{-2}a(1 - r) = \cancel{-2}$$ $$a(1 - r) = 1$$ 5. **Solve the system:** From step 3: $$2ar = -8 \Rightarrow ar = -4$$ From step 4: $$a(1 - r) = 1$$ Express $$a$$ from the second equation: $$a = \frac{1}{1 - r}$$ Substitute into $$ar = -4$$: $$\frac{1}{1 - r} \times r = -4$$ Multiply both sides by $$1 - r$$: $$r = -4(1 - r)$$ $$r = -4 + 4r$$ Bring terms to one side: $$r - 4r = -4$$ $$-3r = -4$$ $$r = \frac{4}{3}$$ 6. **Find $$a$$:** $$a = \frac{1}{1 - r} = \frac{1}{1 - \frac{4}{3}} = \frac{1}{\frac{3}{3} - \frac{4}{3}} = \frac{1}{-\frac{1}{3}} = -3$$ 7. **Final function:** $$f(x) = -3(x + 1)(x - 2)\left(x - \frac{4}{3}\right)$$ 8. **Domain:** The domain of any cubic polynomial is all real numbers: $$\text{Domain} = (-\infty, \infty)$$ 9. **Range:** Since the leading coefficient $$a = -3 < 0$$, the cubic opens downward. The range is also all real numbers: $$\text{Range} = (-\infty, \infty)$$ 10. **Graph sketch description:** - The function crosses the x-axis at $$x = -1$$, $$x = 2$$, and $$x = \frac{4}{3}$$. - The y-intercept is at $$f(0) = -8$$. - The graph passes through $$(1, -2)$$. - The cubic shape with negative leading coefficient means it rises to the left and falls to the right. - Axes are labeled "x" and "y" with arrows indicating positive directions. --- **Answer summary:** a) Remaining zero: $$x = \frac{4}{3}$$ b) Domain: $$(-\infty, \infty)$$, Range: $$(-\infty, \infty)$$ c) Function: $$f(x) = -3(x + 1)(x - 2)\left(x - \frac{4}{3}\right)$$ ---