1. **State the problem:** We are given the polynomial function $$f(x) = -7x^2(x^2 - 3)$$ and need to find: (a) The real zeros and their multiplicities. (b) Whether the graph crosses or touches the x-axis at each zero. (c) The maximum number of turning points. (d) The end behavior of the graph. 2. **Find the real zeros:** Set $$f(x) = 0$$. $$-7x^2(x^2 - 3) = 0$$ This product is zero if either factor is zero: - $$x^2 = 0 \implies x = 0$$ - $$x^2 - 3 = 0 \implies x^2 = 3 \implies x = \pm \sqrt{3}$$ 3. **Determine multiplicities:** - The factor $$x^2$$ corresponds to zero at $$x=0$$ with multiplicity 2. - The factor $$(x^2 - 3)$$ corresponds to zeros at $$x=\pm \sqrt{3}$$ each with multiplicity 1. 4. **Cross or touch at zeros:** - Zeros with even multiplicity (like 2) cause the graph to touch the x-axis and turn around. - Zeros with odd multiplicity (like 1) cause the graph to cross the x-axis. Therefore: - At $$x=0$$ (multiplicity 2), the graph touches the x-axis. - At $$x=\pm \sqrt{3}$$ (multiplicity 1), the graph crosses the x-axis. 5. **Maximum number of turning points:** - The degree of $$f(x)$$ is 4 (since $$x^2 \cdot x^2 = x^4$$). - A polynomial of degree $$n$$ has at most $$n-1$$ turning points. - So, maximum turning points = $$4 - 1 = 3$$. 6. **End behavior:** - For large $$|x|$$, the $$-7x^2(x^2 - 3)$$ behaves like $$-7x^4$$. - Since the leading term is $$-7x^4$$ (degree 4, negative leading coefficient), as $$x \to \pm \infty$$, $$f(x) \to -\infty$$. **Final answers:** (a) Real zeros: $$0$$ (multiplicity 2), $$\sqrt{3}$$ (multiplicity 1), $$-\sqrt{3}$$ (multiplicity 1). (b) At $$x=0$$, graph touches the x-axis; at $$x=\pm \sqrt{3}$$, graph crosses the x-axis. (c) Maximum turning points: 3. (d) End behavior resembles $$-7x^4$$, so $$f(x) \to -\infty$$ as $$x \to \pm \infty$$.