Polynomial Analysis 40469C

1. **State the problem:** We analyze the polynomial function $$f(x) = -2x^3(x + 1)^2(x - 3)$$ to find zeros, multiplicities, intercepts, end behavior, turning points, and sketch the graph. 2. **Find all real zeros and their multiplicities:** - The zeros come from factors set to zero: - $x = 0$ from $x^3$ with multiplicity 3. - $x = -1$ from $(x + 1)^2$ with multiplicity 2. - $x = 3$ from $(x - 3)$ with multiplicity 1. 3. **Behavior at each x-intercept:** - If multiplicity is odd, the graph crosses the x-axis. - If multiplicity is even, the graph touches and turns around. - At $x=0$ (multiplicity 3, odd), graph crosses. - At $x=-1$ (multiplicity 2, even), graph touches. - At $x=3$ (multiplicity 1, odd), graph crosses. 4. **Find the y-intercept:** - Evaluate $f(0)$: $$f(0) = -2 \times 0^3 \times (0 + 1)^2 \times (0 - 3) = 0$$ - So, y-intercept is at $(0,0)$. 5. **Determine the end behavior:** - Degree of polynomial is $3 + 2 + 1 = 6$ (even). - Leading coefficient is negative ($-2$). - For even degree and negative leading coefficient: - As $x \to \infty$, $f(x) \to -\infty$. - As $x \to -\infty$, $f(x) \to -\infty$. 6. **Maximum number of turning points:** - Maximum turning points = degree - 1 = $6 - 1 = 5$. 7. **Sketch a rough graph showing key features:** - Zeros at $x=-1,0,3$ with behaviors as above. - Y-intercept at $(0,0)$. - Ends go down on both sides. Final answers: - Real zeros and multiplicities: $x=0$ (3), $x=-1$ (2), $x=3$ (1). - Behavior at intercepts: crosses at $x=0$ and $x=3$, touches at $x=-1$. - Y-intercept: $(0,0)$. - End behavior: $f(x) \to -\infty$ as $x \to \pm \infty$. - Max turning points: 5.