1. **State the problem:** Given the function $P(x) = x(x - 3)(x + 2)$, find the y-intercept, the x-intercepts, and the end behavior as $x \to \infty$ and $x \to -\infty$. 2. **Find the y-intercept:** The y-intercept occurs when $x=0$. $$P(0) = 0 \times (0 - 3) \times (0 + 2) = 0$$ So, the y-intercept is $0$. 3. **Find the x-intercepts:** The x-intercepts occur when $P(x) = 0$. Set each factor equal to zero: $$x = 0$$ $$x - 3 = 0 \implies x = 3$$ $$x + 2 = 0 \implies x = -2$$ Ordering them from smallest to largest: $$x_1 = -2, \quad x_2 = 0, \quad x_3 = 3$$ 4. **Determine end behavior:** The function is a cubic polynomial with leading term obtained by multiplying the leading terms of each factor: $$x \times x \times x = x^3$$ Since the leading coefficient is positive (1), as $x \to \infty$, $P(x) \to \infty$ (positive infinity). As $x \to -\infty$, since the degree is odd and leading coefficient positive, $P(x) \to -\infty$ (negative infinity). **Final answers:** - y-intercept: $0$ - x-intercepts: $x_1 = -2$, $x_2 = 0$, $x_3 = 3$ - As $x \to \infty$, $y \to +\infty$ - As $x \to -\infty$, $y \to -\infty$