Questions: The polynomial (P(x)) of degree 4 has - a root of multiplicity 2 at (x=1) - a root of multiplicity 1 at (x=0) and at (x=-4) - It goes through the point ((5,576)) Find a formula for (P(x)).

The polynomial (P(x)) of degree 4 has
- a root of multiplicity 2 at (x=1)
- a root of multiplicity 1 at (x=0) and at (x=-4)
- It goes through the point ((5,576))

Find a formula for (P(x)).

Transcript text: The polynomial $P(x)$ of degree 4 has - a root of multiplicity 2 at $x=1$ - a root of multiplicity 1 at $x=0$ and at $x=-4$ - It goes through the point $(5,576)$ Find a formula for $P(x)$. \[ P(x)= \] $\square$

Solution

Solution Steps

Step 1: Construct the Polynomial

Given the roots and their multiplicities, the polynomial can be represented as: $$P(x) = k(x - r)^2(x - s)(x - t)$$ where $k$ is a constant that needs to be determined.

Step 2: Determine the Constant $k$

Using the given point $(u, v)$, we substitute $x = u$ into the polynomial to get: $$P(u) = k(u - r)^2(u - s)(u - t)$$ Solving for $k$ using the equation $P(u) = v$, we find: $$k = \frac{v}{(u - r)^2(u - s)(u - t)} = 0.8$$

Step 3: Formulate the Polynomial

Substituting $k$ back into the polynomial, we get the final form of $P(x)$: $$P(x) = 0.8(x - 1)^2(x - 0)(x + 4)$$