The polynomial $P(x)$ has the following characteristics:

• A root of multiplicity 2 at $x = 4$.
• Roots of multiplicity 1 at $x = 0$ and $x = -3$.
• It passes through the point $(-1, -200)$.

Step 1: Express $P(x)$ in factored form

Given the information about the roots and their multiplicities, the polynomial can be written as: $P(x) = a(x - 4)^2(x)(x + 3)$ where $a$ is a constant to be determined.

Step 2: Use the point $(-1, -200)$

We know that $P(-1) = -200$. Substitute $x = -1$ into the polynomial to solve for $a$:

$P(-1) = a(-1 - 4)^2(-1)(-1 + 3) = -200$

Simplifying the expression inside the parentheses: $(-1 - 4) = -5, \quad (-1 + 3) = 2$ $P(-1) = a(-5)^2(-1)(2) = -200$

Now calculate the powers and products: $P(-1) = a(25)(-1)(2) = -200$ $P(-1) = -50a = -200$

Step 3: Solve for $a$

$a = \frac{-200}{-50} = 4$

Step 4: Write the final formula

Now substitute $a = 4$ back into the factored form of the polynomial:

$P(x) = 4(x - 4)^2 x (x + 3)$

This is the formula for the polynomial $P(x)$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do we know the degree of the polynomial based on the roots and their multiplicities?
2. What does it mean for a root to have multiplicity 2 versus multiplicity 1?
3. How can you find the constant $a$ if a point is not given?
4. How do changes in the coefficient $a$ affect the graph of the polynomial?
5. How would the polynomial change if another point on the graph was given instead of $(-1, -200)$?

Tip: Always verify your solution by substituting points given in the problem into your equation.