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The problem asks for the formula of a polynomial \( P(x) \) of degree 3, with specific roots and conditions:

- A root of multiplicity 2 at \( x = 3 \),
- A root of multiplicity 1 at \( x = -3 \),
- A given y-intercept of \( y = -2.7 \) (i.e., \( P(0) = -2.7 \)).

### Step 1: Set up the general form of the polynomial

Since there is a root of multiplicity 2 at \( x = 3 \), the factor corresponding to this root is \( (x - 3)^2 \).

For the root at \( x = -3 \) (multiplicity 1), the factor is \( (x + 3) \).

Thus, the general form of the polynomial is:

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

where \( a \) is a constant that we need to determine.

### Step 2: Use the y-intercept to find \( a \)

We are told that the y-intercept is \( y = -2.7 \), meaning \( P(0) = -2.7 \). Substituting \( x = 0 \) into the equation:

\[
P(0) = a(0 - 3)^2(0 + 3)
\]
\[
P(0) = a(9)(3) = 27a
\]
\[
-2.7 = 27a
\]

Solving for \( a \):

\[
a = \frac{-2.7}{27} = -0.1
\]

### Step 3: Final formula for \( P(x) \)

Now that we know \( a = -0.1 \), the polynomial is:

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

This is the required polynomial.

Would you like more details on any part of this? Here are some related questions:

1. What does the multiplicity of a root mean in a polynomial function?
2. How does the y-intercept affect the scaling factor of a polynomial?
3. How would the polynomial change if the root at \( x = -3 \) had multiplicity 2 instead?
4. Can you explain why the degree of the polynomial matches the sum of the root multiplicities?
5. How can the polynomial be expanded into its standard form?

**Tip**: The multiplicity of a root describes how many times the root repeats in the factorization of the polynomial. It affects the behavior of the graph at that root, causing the graph to touch the x-axis but not cross it when the multiplicity is even.

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x=3 and a root of multiplicity 1 at x=-3. The y-intercept is y=-2.7. Find a formula for P(x).

Learn how to find the formula of a degree 3 polynomial with a root of multiplicity 2 at x=3 and a root of multiplicity 1 at x=-3. The polynomial has a y-intercept of -2.7. This step-by-step guide covers the core concepts of polynomial roots and y-intercepts for high school algebra students.

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