A polynomial of degree 8, P⁡(x) has leading coefficient 1, and has roots of multiplicity 1 at x=0, multiplicity 3 at x=-4, and multiplicity 4 at x=-2.

Given the conditions for the polynomial \( P(x) \):

- The polynomial is of degree 8.
- The leading coefficient is 1.
- There is a root of multiplicity 1 at \( x = 0 \).
- There is a root of multiplicity 3 at \( x = -4 \).
- There is a root of multiplicity 4 at \( x = -2 \).

We can express \( P(x) \) in its factored form as follows:

\[
P(x) = a(x - 0)^1(x + 4)^3(x + 2)^4
\]

Given that the leading coefficient is 1, the constant \( a = 1 \). Thus, the polynomial simplifies to:

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

This is the polynomial \( P(x) \) that satisfies all the given conditions.

Would you like more details or have any questions?

Here are some related questions:

1. How can the degree of a polynomial be determined by its roots?
2. What is the significance of the leading coefficient in a polynomial?
3. How do you find the multiplicity of a root in a polynomial?
4. How would the polynomial change if the leading coefficient was different?
5. Can you graph the polynomial to verify its roots and multiplicities?

**Tip:** Always check that the sum of the multiplicities of the roots matches the degree of the polynomial to ensure accuracy.

Learn how to construct a degree 8 polynomial with a leading coefficient of 1, having roots of multiplicity 1 at x = 0, multiplicity 3 at x = -4, and multiplicity 4 at x = -2. Detailed step-by-step solution and polynomial formulation provided.

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