1. **Problem statement:** Find the x- and y-intercepts of the polynomial function $$f(x) = x^2(x - 5)(x^2 + 2)$$. 2. **Recall definitions:** - The **x-intercepts** are the values of $x$ where $f(x) = 0$. - The **y-intercept** is the value of $f(0)$. 3. **Find x-intercepts:** Set $$f(x) = 0$$: $$x^2(x - 5)(x^2 + 2) = 0$$ 4. **Solve each factor:** - $$x^2 = 0 \implies x = 0$$ - $$x - 5 = 0 \implies x = 5$$ - $$x^2 + 2 = 0 \implies x^2 = -2$$ which has no real solutions. 5. **Therefore, the real x-intercepts are:** $$x = 0, 5$$ 6. **Find y-intercept:** Evaluate $$f(0)$$: $$f(0) = 0^2(0 - 5)(0^2 + 2) = 0$$ 7. **Summary:** - x-intercepts: $$0, 5$$ - y-intercept: $$0$$