1. **State the problem:** We need to factor the function $$f(x) = -27x^5 - 18x^4 - 3x^3$$ to find all its x-intercepts. 2. **Recall the rule for x-intercepts:** The x-intercepts occur where $$f(x) = 0$$. To find these, we factor the function and solve for $$x$$. 3. **Factor out the greatest common factor (GCF):** The coefficients are -27, -18, and -3. The GCF of 27, 18, and 3 is 3, and all terms have at least $$x^3$$. So, factor out $$-3x^3$$: $$f(x) = -3x^3(9x^2 + 6x + 1)$$ 4. **Factor the quadratic inside the parentheses:** We want to factor $$9x^2 + 6x + 1$$. Check if it factors nicely: $$9x^2 + 6x + 1 = (3x + 1)^2$$ 5. **Rewrite the factored form:** $$f(x) = -3x^3(3x + 1)^2$$ 6. **Find the x-intercepts by setting each factor equal to zero:** - $$-3x^3 = 0 \\ \Rightarrow x^3 = 0 \\ \Rightarrow x = 0$$ - $$(3x + 1)^2 = 0 \\ \Rightarrow 3x + 1 = 0 \\ \Rightarrow x = -\frac{1}{3}$$ 7. **Final answer:** The x-intercepts are $$x = 0$$ and $$x = -\frac{1}{3}$$. These are the points where the graph crosses or touches the x-axis.