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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?

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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.

Write an equation for a polynomial function whose graph intersects the horizontal axis at -14, -5, and 4 and has a vertical intercept at (0, -56).

Learn how to find the equation of a polynomial function with x-intercepts at -14, -5, and 4 and a vertical intercept at (0, -56). This problem involves key algebraic concepts such as polynomial functions, x-intercepts, and the Factor Theorem. Suitable for students in Grades 9-12.

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