To find the formula for $P(x)$, let's carefully use the information from the problem.

Step 1: Structure of the Polynomial

The given polynomial $P(x)$ is of degree 3.

• It has a root with multiplicity 2 at $x = 2$.
• It has a root with multiplicity 1 at $x = -1$.

Thus, the general form of $P(x)$ will be: $P(x) = a \cdot (x - 2)^2 \cdot (x + 1),$ where $a$ is a constant we need to determine.

Step 2: Use the $y$-intercept

We are told that the y-intercept is $y = -2$.
At the y-intercept, $x = 0$, so we substitute $x = 0$ into the polynomial.

$P(0) = a \cdot (0 - 2)^2 \cdot (0 + 1).$ $P(0) = a \cdot 4 \cdot 1 = 4a.$ Since $P(0) = -2$, we can set up the equation: $4a = -2.$ Solving for $a$: $a = \frac{-2}{4} = -\frac{1}{2}.$

Step 3: Write the Final Formula

Now that we know $a = -\frac{1}{2}$, the formula for the polynomial is:

$P(x) = -\frac{1}{2} \cdot (x - 2)^2 \cdot (x + 1).$

Would you like further details, such as graphing the polynomial or checking its behavior?

Related Questions:

1. How do we find the roots of higher-degree polynomials?
2. What does it mean for a root to have multiplicity greater than 1?
3. How can we confirm that the degree of a polynomial matches the given properties?
4. What happens to the graph of a polynomial at roots with even and odd multiplicities?
5. How can polynomial functions be factored completely?

Tip: Roots with even multiplicities cause the graph to touch the x-axis without crossing it, while roots with odd multiplicities cause the graph to cross the x-axis.