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The problem asks for the polynomial \( P(x) \) of degree 3, given the following conditions:

1. A root of multiplicity 2 at \( x = 1 \),
2. A root of multiplicity 1 at \( x = -1 \),
3. The y-intercept is \( y = -0.3 \), meaning \( P(0) = -0.3 \).

### Step-by-step solution:

1. **Express the polynomial with the given roots:**
   - Since there is a root of multiplicity 2 at \( x = 1 \), one factor of \( P(x) \) is \( (x - 1)^2 \).
   - Since there is a root of multiplicity 1 at \( x = -1 \), another factor is \( (x + 1) \).

Thus, the polynomial can be written as:
\[
P(x) = a(x - 1)^2(x + 1)
\]
where \( a \) is a constant to be determined.

2. **Use the y-intercept condition:**
   - We are told that \( P(0) = -0.3 \). Plug \( x = 0 \) into the polynomial:
   \[
   P(0) = a(0 - 1)^2(0 + 1) = a(1)(1) = a
   \]
   Therefore, \( a = -0.3 \).

3. **Write the final polynomial:**
   Substituting \( a = -0.3 \) into the polynomial, we get:
   \[
   P(x) = -0.3(x - 1)^2(x + 1)
   \]

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

Would you like more details or explanations?

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Here are some related questions to deepen your understanding:

1. What does it mean for a root to have a "multiplicity" of 2 in a polynomial?
2. How would the graph of this polynomial behave at \( x = 1 \) and \( x = -1 \)?
3. How would the polynomial change if the y-intercept were a different value?
4. What would happen if there were a third root with a different multiplicity?
5. How can we verify that the polynomial satisfies all given conditions?

**Tip:** Always check the degree of the polynomial to ensure it matches the number of roots and their multiplicities!

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 1 and a root of multiplicity 1 at x = -1. The y-intercept is y = -0.3. Find a formula for P(x).

This problem involves finding the polynomial P(x) of degree 3 given roots of multiplicity at x=1 and x=-1, along with a y-intercept of y = -0.3. Learn how to factor and use given conditions to derive the correct polynomial formula.

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