1. **Problem Statement:** We see a polynomial curve $p(x)$ that crosses the x-axis before $x=1$, has a local maximum between $x=1$ and $x=2$, and then a local minimum near $x=2$. After that, the curve rises sharply upward. We want a polynomial that behaves like this. 2. **Analyzing the behavior:** The curve has: - At least one root slightly before $x=1$, so a root slightly less than 1. - A local maximum between $1$ and $2$, so the derivative should be zero there. - A local minimum near $x=2$. - End behavior rising sharply upward after $x=2$, so degree of the polynomial is even and leading coefficient is positive or degree is odd and leading coefficient is positive causing sharp rise at the right. 3. **Constructing a polynomial:** A cubic polynomial can have one local minimum and one local maximum. The curve crosses the x-axis once. Let's try a cubic polynomial with roots near $0.8$, $1.8$, and one more root possibly not shown or complex. 4. **Example polynomial:** Let’s try the form $$p(x) = (x - 0.8)(x - 1.8)^2$$ - It has a root at $x=0.8$ (single root, so crosses x-axis) - A double root at $x=1.8$ (touches x-axis and turns there), but the graph description says it crosses near $x=1$ only once and then has a max and min, so 5. Instead, consider a cubic polynomial with three real roots with local min and max points: $$p(x) = (x - a)(x - b)(x - c)$$ where $a < 1 < b < 2 < c$ - The graph crosses x-axis near $x=1$ and rises sharply at x=2 (which could be near $c$). 6. **Simplest guess:** A cubic polynomial with roots at $x=0.9$, $x=1.5$, and $x=2.2$: $$p(x) = (x - 0.9)(x - 1.5)(x - 2.2)$$ This polynomial will: - Cross x-axis slightly before 1 - Have a local max and min between the roots - Rise sharply at $x=2$, consistent with behavior described **Final answer:** $$p(x) = (x - 0.9)(x - 1.5)(x - 2.2)$$ This matches the graph description well.