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To find the formula for \( P(x) \), let's carefully use the information from the problem.

### Step 1: Structure of the Polynomial
The given polynomial \( P(x) \) is of degree 3.  
- It has a **root with multiplicity 2** at \( x = 2 \).  
- It has a **root with multiplicity 1** at \( x = -1 \).  

Thus, the general form of \( P(x) \) will be:
\[
P(x) = a \cdot (x - 2)^2 \cdot (x + 1),
\]
where \( a \) is a constant we need to determine.

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### Step 2: Use the \( y \)-intercept
We are told that the **y-intercept** is \( y = -2 \).  
At the y-intercept, \( x = 0 \), so we substitute \( x = 0 \) into the polynomial.

\[
P(0) = a \cdot (0 - 2)^2 \cdot (0 + 1).
\]
\[
P(0) = a \cdot 4 \cdot 1 = 4a.
\]
Since \( P(0) = -2 \), we can set up the equation:
\[
4a = -2.
\]
Solving for \( a \):
\[
a = \frac{-2}{4} = -\frac{1}{2}.
\]

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### Step 3: Write the Final Formula
Now that we know \( a = -\frac{1}{2} \), the formula for the polynomial is:

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

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Would you like further details, such as graphing the polynomial or checking its behavior?

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### Related Questions:
1. How do we find the roots of higher-degree polynomials?
2. What does it mean for a root to have multiplicity greater than 1?
3. How can we confirm that the degree of a polynomial matches the given properties?
4. What happens to the graph of a polynomial at roots with even and odd multiplicities?
5. How can polynomial functions be factored completely?

**Tip:** Roots with even multiplicities cause the graph to touch the x-axis without crossing it, while roots with odd multiplicities cause the graph to cross the x-axis.

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 2 and a root of multiplicity 1 at x = -1. The y-intercept is y = -2. Find a formula for P(x).

Find the polynomial P(x) of degree 3 with roots at x = 2 (multiplicity 2) and x = -1 (multiplicity 1), and a y-intercept of y = -2. This problem involves the concepts of roots, multiplicities, and polynomial structure, suitable for students in Grades 9-12.

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