To create a polynomial $p(x)$ that satisfies the given conditions, we need to consider each requirement step by step:

1. Degree of the polynomial is 5: This means the polynomial will have the general form $p(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)(x - r_5)$ where $r_1, r_2, r_3, r_4, r_5$ are the roots of the polynomial, and $a$ is the leading coefficient.

2. As $x \to -\infty$, $p(x) \to \infty$: For this behavior, the leading coefficient $a$ must be negative since a polynomial of odd degree (5 in this case) will go to opposite infinities on either end. Specifically, if $a < 0$, then $p(x)$ will tend to $\infty$ as $x \to -\infty$.

3. The point $(-1, 0)$ yields a local maximum: This implies that $x = -1$ is both a root and a critical point where the derivative changes sign from positive to negative. Thus, $(x + 1)^2$ should be a factor of the polynomial, ensuring that $x = -1$ is a double root and hence a local maximum.

4. The point $(9, 0)$ is one of the x-intercepts: This implies $x = 9$ is a root of the polynomial.

Given these conditions, the polynomial can be expressed as: $p(x) = -a(x + 1)^2(x - 9)(x - r_1)(x - r_2)$ where $r_1$ and $r_2$ are the remaining roots.

To fully specify $p(x)$, we could assume additional roots or constraints, but based on the information given, a simple version with the remaining roots at $x = 0$ and $x = 2$ (just as an example) could be: $p(x) = -1(x + 1)^2(x - 9)x(x - 2)$

This polynomial has the following properties:

• Degree 5 (since it is the product of five factors).
• As $x \to -\infty$, $p(x) \to \infty$ because of the negative leading coefficient.
• $x = -1$ is a double root, ensuring a local maximum at this point.
• $x = 9$ is an x-intercept.

Final Polynomial:

$p(x) = -(x + 1)^2(x - 9)x(x - 2)$

Do you want any more details or have any questions?

Related Questions:

1. How does the sign of the leading coefficient affect the end behavior of a polynomial?
2. What is the significance of a double root in a polynomial?
3. How can you determine the number of local maxima and minima in a polynomial?
4. What happens to the graph of a polynomial at a point where it has an odd multiplicity root?
5. How do we derive the derivative of a polynomial and use it to find critical points?

Tip:

When constructing polynomials with specific behaviors, it's crucial to consider the multiplicity of roots and the sign of the leading coefficient, as these significantly influence the shape and behavior of the graph.