The function $p(x) = x(x-5)(x+4)$ is a polynomial, and we can calculate its x-intercepts and y-intercept as follows:

1. X-Intercepts

The x-intercepts occur when $p(x) = 0$. Set the equation equal to zero: $x(x-5)(x+4) = 0$

This equation is satisfied if any of the factors is zero:

• $x = 0$
• $x - 5 = 0$ → $x = 5$
• $x + 4 = 0$ → $x = -4$

Thus, the x-intercepts are: $x = 0, x = 5, \text{ and } x = -4$

These correspond to the points: $(0, 0), (5, 0), (-4, 0)$

2. Y-Intercept

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the function: $p(0) = 0(0-5)(0+4) = 0$

Thus, the y-intercept is: $(0, 0)$

Summary:

• X-Intercepts: $(0, 0), (5, 0), (-4, 0)$
• Y-Intercept: $(0, 0)$

Would you like a graph or more explanation about how these points affect the graph of the function?

Expanding Questions:

1. What is the shape of the graph of this cubic polynomial?
2. How does the sign of the leading coefficient affect the behavior of the function?
3. What are the turning points of the graph, and how can they be determined?
4. How do the x-intercepts relate to the factors of the polynomial?
5. What is the end behavior of the function $p(x)$ as $x \to \pm \infty$?

Tip:

For polynomials, the number of x-intercepts corresponds to the number of real roots of the equation $p(x) = 0$. The degree of the polynomial determines the maximum possible number of x-intercepts.