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We are given a polynomial \( P(x) \) of degree 4, with the following characteristics:

- A root of multiplicity 2 at \( x = 1 \),
- Roots of multiplicity 1 at \( x = 0 \) and \( x = -4 \),
- The polynomial passes through the point \( (5, 576) \).

### Step 1: General Form of the Polynomial

Since the polynomial has roots at \( x = 1 \) (with multiplicity 2), \( x = 0 \), and \( x = -4 \), we can express the polynomial as:

\[
P(x) = a(x - 1)^2 x (x + 4)
\]

Here, \( a \) is a constant that we need to determine.

### Step 2: Use the Point \( (5, 576) \)

To find the value of \( a \), substitute \( x = 5 \) and \( P(x) = 576 \) into the polynomial equation:

\[
576 = a(5 - 1)^2 (5)(5 + 4)
\]

Simplifying the terms:

\[
576 = a(4^2)(5)(9)
\]
\[
576 = a(16)(5)(9)
\]
\[
576 = a \cdot 720
\]

Now, solve for \( a \):

\[
a = \frac{576}{720} = \frac{4}{5}
\]

### Step 3: Final Formula for \( P(x) \)

Thus, the polynomial is:

\[
P(x) = \frac{4}{5}(x - 1)^2 x (x + 4)
\]

This is the required formula for \( P(x) \).

Would you like further clarification on any step?

### Related Questions:
1. How do we determine the degree of a polynomial based on its roots and multiplicities?
2. What is the significance of multiplicity in polynomial roots?
3. Can a polynomial have multiple points where it passes through a certain \( y \)-value?
4. How would the formula change if the point was \( (5, 100) \) instead of \( (5, 576) \)?
5. How can you factor a polynomial when given roots with varying multiplicities?

### Tip:
When a polynomial has roots with multiplicities, the root will "flatten" near that point, meaning the graph touches the x-axis but does not cross it.

The polynomial of degree 4, P(x), has a root of multiplicity 2 at x = 1 and roots of multiplicity 1 at x = 0 and x = -4. It goes through the point (5, 576). Find a formula for P(x).

Find the formula for a degree 4 polynomial, P(x), with roots of multiplicity 2 at x = 1, multiplicity 1 at x = 0 and x = -4, and passing through the point (5, 576). Includes step-by-step calculations.

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