1. **Problem:** Analyze the polynomial function $$y = (x - 3)^2 (x + 5)^2$$. 2. **Degree:** The degree is the sum of the exponents: $$2 + 2 = 4$$. 3. **Zeros and Types:** - Zero at $$x=3$$ with multiplicity 2 (even multiplicity means the graph touches the x-axis and turns around). - Zero at $$x=-5$$ with multiplicity 2 (same behavior). 4. **Y-intercept:** Substitute $$x=0$$: $$y = (0 - 3)^2 (0 + 5)^2 = 9 \times 25 = 225$$. 5. **End Behavior:** Since the leading term is positive and degree is even, as $$x \to \pm \infty$$, $$y \to +\infty$$. --- 1. **Problem:** Analyze the polynomial function $$y = - (x + 3)(x - 6)(x + 5)(x - 2)$$. 2. **Degree:** There are four linear factors, so degree is 4. 3. **Zeros and Types:** - Zero at $$x=-3$$ (multiplicity 1, crosses x-axis). - Zero at $$x=6$$ (multiplicity 1, crosses x-axis). - Zero at $$x=-5$$ (multiplicity 1, crosses x-axis). - Zero at $$x=2$$ (multiplicity 1, crosses x-axis). 4. **Y-intercept:** Substitute $$x=0$$: $$y = - (0 + 3)(0 - 6)(0 + 5)(0 - 2) = - (3)(-6)(5)(-2) = - (3 \times -6 \times 5 \times -2)$$ Calculate stepwise: $$3 \times -6 = -18$$ $$-18 \times 5 = -90$$ $$-90 \times -2 = 180$$ So, $$y = - (180) = -180$$. 5. **End Behavior:** Leading coefficient is negative and degree is even, so as $$x \to \pm \infty$$, $$y \to -\infty$$. --- Since the user requested only end behavior for the first two problems, for the remaining four problems, we provide degree, zeros and types, and y-intercept only. --- 3. **Problem:** $$y = (x + 3)(x - 2)(x - 4)$$ - Degree: 3 - Zeros: $$x=-3, 2, 4$$ all multiplicity 1 (cross x-axis) - Y-intercept: $$y = (0+3)(0-2)(0-4) = 3 \times (-2) \times (-4) = 24$$ 4. **Problem:** $$y = (x + 4)(x - 5)^2$$ - Degree: 3 - Zeros: $$x=-4$$ multiplicity 1 (cross), $$x=5$$ multiplicity 2 (touch and turn) - Y-intercept: $$y = (0+4)(0-5)^2 = 4 \times 25 = 100$$ 5. **Problem:** $$y = - (x - 3)(x + 5)^2$$ - Degree: 3 - Zeros: $$x=3$$ multiplicity 1 (cross), $$x=-5$$ multiplicity 2 (touch and turn) - Y-intercept: $$y = - (0 - 3)(0 + 5)^2 = - (-3)(25) = 75$$ 6. **Problem:** $$y = x(x + 4)(x - 5)^2$$ - Degree: 4 - Zeros: $$x=0$$ multiplicity 1 (cross), $$x=-4$$ multiplicity 1 (cross), $$x=5$$ multiplicity 2 (touch and turn) - Y-intercept: $$y = 0 \times 4 \times 25 = 0$$