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:

• $-0.1(x - 1)^2(x + 3)$

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.