The problem asks to find the equation of a polynomial function with specific properties:

• The graph intersects the horizontal axis at $x = -14$, $x = -5$, and $x = 4$. These are the x-intercepts of the polynomial, meaning the roots of the equation.
• It has a vertical intercept at $(0, -56)$, meaning the value of the polynomial at $x = 0$ is $-56$.

The general form of a polynomial with roots $x = -14$, $x = -5$, and $x = 4$ can be written as: $f(x) = a(x + 14)(x + 5)(x - 4)$ where $a$ is a constant that will be determined by the vertical intercept.

To find $a$, we substitute $x = 0$ and $f(0) = -56$: $-56 = a(0 + 14)(0 + 5)(0 - 4)$ $-56 = a(14)(5)(-4)$ $-56 = a(-280)$ $a = \frac{-56}{-280} = 0.2$

Thus, the equation of the polynomial is: $f(x) = 0.2(x + 14)(x + 5)(x - 4)$

This corresponds to the third option in the given choices:

• $0.2(x + 14)(x + 5)(x - 4)$

Would you like more details on this, or have any questions?

Here are five related questions:

1. How do you find the degree of a polynomial given its roots?
2. What is the significance of the constant $a$ in the polynomial equation?
3. How do you determine the x-intercepts of a polynomial from its equation?
4. How does the value of $f(0)$ determine the vertical intercept?
5. Can a polynomial have multiple vertical intercepts?

Tip: Always use the x-intercepts to directly construct the factors of a polynomial equation, and then adjust the constant based on any given additional points.