The polynomial of degree 33, P(x)P(x), has a root of multiplicity 22 at x=3x=3 and a root of multiplicity 11 at x=−3x=-3. The yy-intercept is y=−16.2y=-16.2.

Find a formula for P(x)P(x).
P(x)=P(x)=

Given the information in the problem, we can construct the polynomial step by step.

### Step 1: General form of the polynomial
We are told that the polynomial \( P(x) \) has:
- A root of multiplicity 22 at \( x = 3 \).
- A root of multiplicity 11 at \( x = -3 \).

Thus, the polynomial can be written as:

\[
P(x) = a(x - 3)^{22}(x + 3)^{11}
\]

Where \( a \) is a constant that we will determine using the given \( y \)-intercept.

### Step 2: Use the \( y \)-intercept to find \( a \)
We are also told that the \( y \)-intercept is \( y = -16.2 \), which occurs when \( x = 0 \).

Substitute \( x = 0 \) into the polynomial:

\[
P(0) = a(0 - 3)^{22}(0 + 3)^{11}
\]
\[
P(0) = a(-3)^{22}(3)^{11}
\]

Since \( (-3)^{22} = 3^{22} \), we can rewrite this as:

\[
P(0) = a(3^{22})(3^{11}) = a(3^{33})
\]

We are given that \( P(0) = -16.2 \), so:

\[
-16.2 = a(3^{33})
\]

### Step 3: Solve for \( a \)
Now, solve for \( a \):

\[
a = \frac{-16.2}{3^{33}}
\]

This gives us the value of \( a \).

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

\[
P(x) = \frac{-16.2}{3^{33}}(x - 3)^{22}(x + 3)^{11}
\]

Would you like further details or clarifications on any steps? 

Here are 5 related questions:
1. What is a polynomial's degree and how is it determined?
2. How do multiplicities of roots affect the shape of the graph?
3. How does the \( y \)-intercept of a polynomial help in finding unknown constants?
4. Can the multiplicity of roots be any number, and how does it impact the curve at those points?
5. Why are some polynomials factorizable and others not?

**Tip:** To find the constant \( a \), using known points on the curve like intercepts is a common strategy in polynomial problems.

The polynomial of degree 33, P(x), has a root of multiplicity 22 at x = 3 and a root of multiplicity 11 at x = -3. The y-intercept is y = -16.2. Find a formula for P(x).

Discover how to find the formula for a polynomial of degree 33 with roots of multiplicity 22 and 11 at x=3 and x=-3, respectively. This solution involves calculating the constant using the y-intercept of -16.2. Suitable for students in Grades 10-12.

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