Question: For the polynomial function $f(x) = - x^2 (x - 3)(x + 5)$, answer parts a through e.
a. Use the Leading Coefficient Test to determine the graph's end behavior.
A. The graph of $f(x)$ rises to the left and falls to the right.
B. The graph of $f(x)$ falls to the left and falls to the right.
C. The graph of $f(x)$ rises to the left and rises to the right.
D. The graph of $f(x)$ falls to the left and rises to the right.
b. Find the x-intercept(s).
The x-intercept(s) is/are
(Type an integer or a decimal. Use a comma to separate answers as needed.)
Graph shape: the polynomial graph is a quartic with negative leading coefficient, so its end behavior falls to the left and falls to the right; x-intercepts at $x = -5$, $0$, $3$ with $x = 0$ having multiplicity 2. position_hint=centerplot this on a graph
1. **State the problem:** We analyze the polynomial function $$f(x) = - x^2 (x - 3)(x + 5)$$ to determine its end behavior and find its x-intercepts.
2. **Determine the degree and leading coefficient:**
- Expand the factors to find the degree:
$$f(x) = - x^2 (x - 3)(x + 5) = - x^2 (x^2 + 5x - 3x - 15) = - x^2 (x^2 + 2x - 15)$$
- Multiply out:
$$f(x) = - (x^2)(x^2 + 2x - 15) = - (x^4 + 2x^3 - 15x^2) = -x^4 - 2x^3 + 15x^2$$
- The degree is $$4$$ (highest power of $$x$$).
- The leading coefficient is $$-1$$ (coefficient of $$x^4$$).
3. **Use the Leading Coefficient Test for end behavior:**
- For even degree $$n=4$$ and negative leading coefficient $$-1$$:
- As $$x \to \infty$$, $$f(x) \to -\infty$$ (falls to the right).
- As $$x \to -\infty$$, $$f(x) \to -\infty$$ (falls to the left).
4. **Answer for part a:** The graph falls to the left and falls to the right.
5. **Find the x-intercepts:**
- Set $$f(x) = 0$$:
$$- x^2 (x - 3)(x + 5) = 0$$
- This equals zero when any factor is zero:
- $$x^2 = 0 \implies x = 0$$ (multiplicity 2)
- $$x - 3 = 0 \implies x = 3$$
- $$x + 5 = 0 \implies x = -5$$
6. **Answer for part b:** The x-intercepts are $$x = -5, 0, 3$$ with $$x=0$$ having multiplicity 2.
7. **Summary:**
- End behavior: falls to the left and falls to the right.
- X-intercepts: $$-5, 0, 3$$ (with multiplicity 2 at $$0$$).