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The question asks for the formula of a degree-3 polynomial, \( P(x) \), with the following conditions:

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

### Step 1: General Form of the Polynomial

The polynomial must have the following factors based on the root conditions:

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

- The factor \( (x - 1)^2 \) accounts for the root at \( x = 1 \) with multiplicity 2.
- The factor \( (x + 3) \) accounts for the root at \( x = -3 \) with multiplicity 1.

### Step 2: Use the Y-intercept Condition

The y-intercept occurs when \( x = 0 \). So, to find the constant \( a \), substitute \( x = 0 \) and set \( P(0) = -0.3 \):

\[
P(0) = a(0 - 1)^2(0 + 3) = -0.3
\]

\[
P(0) = a(1)^2(3) = 3a
\]

\[
3a = -0.3 \implies a = \frac{-0.3}{3} = -0.1
\]

### Step 3: Final Polynomial

Thus, the formula for \( P(x) \) is:

\[
P(x) = -0.1(x - 1)^2(x + 3)
\]

### Step 4: Check the Answer Options

We now check the answer choices provided:

- \( 0.1(x + 1)^2(x - 3) \): Incorrect due to wrong signs on both factors.
- \( -0.1(x - 1)^2(x + 3) \): Correct.
- \( \frac{1}{10}(x + 1)^2(x - 3) \): Incorrect due to wrong signs on both factors.
- \( \frac{1}{10}(x - 1)^2(x + 3) \): Incorrect coefficient (it should be negative).
- \( (x + 1)^2(x - 3) \): Incorrect due to wrong signs and missing constant.
- \( (x - 1)^2(x + 3) \): Incorrect as the constant is missing.

### Correct Answer:
- \( -0.1(x - 1)^2(x + 3) \)

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Would you like more details on the process? Here are 5 related questions:

1. How do we determine the multiplicity of roots in a polynomial?
2. What is the significance of the y-intercept when solving for a polynomial?
3. How would the formula change if the y-intercept were different?
4. What happens to the polynomial if a root has a higher multiplicity?
5. Can this method be extended to polynomials of higher degrees?

**Tip:** When solving for polynomials with given roots, always account for multiplicities by using powers in the factors corresponding to each root.

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

Solve a degree-3 polynomial problem with a root at x = 1 with multiplicity 2 and a root at x = -3 with multiplicity 1. Learn how to apply the y-intercept condition (y = -0.3) to find the correct polynomial formula.

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