The polynomial of degree 4,  
P
(
x
)
  has a root of multiplicity 2 at  
x
=
4
  and roots of multiplicity 1 at  
x
=
0
  and  
x
=
−
3
 . The graph passes through the point  
(
−
1
,
−
200
)
 .

Find a formula for  
P
(
x
)
 .

  
P
(
x
)
=

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.

The polynomial of degree 4, P(x) has a root of multiplicity 2 at x = 4 and roots of multiplicity 1 at x = 0 and x = -3. The graph passes through the point (-1, -200). Find a formula for P(x).

Learn how to find the formula for a degree 4 polynomial P(x) with specific roots and multiplicities, including a root of multiplicity 2 at x = 4, and roots of multiplicity 1 at x = 0 and x = -3. The polynomial passes through the point (-1, -200). Step-by-step solution for high school students in Grades 10-12.

Math Problem Statement

The polynomial of degree 4,
P ( x ) has a root of multiplicity 2 at
x

4 and roots of multiplicity 1 at
x

0 and
x

P ( x )

Solution

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