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To find a possible formula for the polynomial \( P(x) \), we need to consider the given conditions:

1. **Degree of the polynomial**: \( P(x) \) is of degree 5.
2. **Leading coefficient**: The leading coefficient is 1.
3. **Roots and their multiplicities**:
   - \( x = 3 \) has multiplicity 2.
   - \( x = 0 \) has multiplicity 2.
   - \( x = -4 \) has multiplicity 1.

Since each root’s multiplicity contributes to the degree, we can construct \( P(x) \) as follows:

\[
P(x) = (x - 3)^2 \cdot x^2 \cdot (x + 4)
\]

Expanding this polynomial isn’t necessary unless specifically required, as the factored form fulfills the conditions of the problem. Given the leading coefficient is 1, this form is correct without additional scaling.

So, the answer is:

\[
P(x) = (x - 3)^2 \cdot x^2 \cdot (x + 4)
\]

Would you like further details on any specific step, or do you have any questions?

Here are some related questions you might find helpful:

1. How would the polynomial change if the leading coefficient was not 1?
2. What happens to the polynomial if we change the multiplicity of \( x = 0 \) from 2 to 3?
3. How can we verify that this polynomial is indeed of degree 5?
4. What would be the effect on the polynomial if we add another root with a different multiplicity?
5. How would we factor a polynomial if given only its expanded form?

**Tip:** When constructing polynomials with specific roots and multiplicities, always verify that the sum of the multiplicities matches the polynomial's degree.

The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x = 3 and x = 0, and a root of multiplicity 1 at x = -4. Find a possible formula for P(x).

Learn how to find a degree 5 polynomial with specified roots and multiplicities. This example problem provides a step-by-step solution to constructing the polynomial P(x) with roots at x = 3, x = 0, and x = -4, including multiplicities. Suitable for high school students in Grades 10-12.

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