https://file.aiquickdraw.com/mathbot/file/1726464234048rk7yospr.jpg

The problem asks to find the polynomial \( P(x) \) of degree 4, which has the following characteristics:

- A root of multiplicity 2 at \( x = 3 \).
- Roots of multiplicity 1 at \( x = 0 \) and \( x = -1 \).
- The polynomial passes through the point \( (5, 48) \).

### Step 1: Form the general structure of the polynomial

The root structure tells us the factors of the polynomial:
- A root at \( x = 3 \) with multiplicity 2 gives the factor \( (x - 3)^2 \).
- A root at \( x = 0 \) gives the factor \( x \).
- A root at \( x = -1 \) gives the factor \( (x + 1) \).

Therefore, the general form of the polynomial is:

\[
P(x) = a(x - 3)^2 x (x + 1)
\]

Where \( a \) is a constant that needs to be determined.

### Step 2: Use the point \( (5, 48) \)

We are given that the polynomial passes through the point \( (5, 48) \), meaning when \( x = 5 \), \( P(x) = 48 \). Substituting \( x = 5 \) into the polynomial:

\[
48 = a(5 - 3)^2 (5)(5 + 1)
\]

Simplifying:

\[
48 = a(2)^2 (5)(6)
\]
\[
48 = a(4)(30)
\]
\[
48 = 120a
\]

Solving for \( a \):

\[
a = \frac{48}{120} = \frac{2}{5}
\]

### Step 3: Write the final polynomial

Now that we know \( a = \frac{2}{5} \), we can write the final form of the polynomial:

\[
P(x) = \frac{2}{5}(x - 3)^2 x (x + 1)
\]

This is the required polynomial of degree 4.

---

Would you like a more detailed explanation or have any further questions?

Here are some related questions to expand the topic:
1. How do you calculate the multiplicity of a root in a polynomial?
2. What happens if a polynomial has complex roots? How is the form affected?
3. How would the polynomial change if it passed through a different point, say \( (2, 16) \)?
4. What is the significance of the constant \( a \) in determining the shape of the polynomial?
5. How would you handle a polynomial with more than one double root?

**Tip:** When determining the form of a polynomial, always start by identifying its roots and their multiplicities, then use any extra conditions (like passing through a point) to find the constant factor.

The polynomial of degree 4, P(x), has a root of multiplicity 2 at x = 3 and roots of multiplicity 1 at x = 0 and x = -1. It goes through the point (5, 48).

This solution explains how to find a degree 4 polynomial with roots of multiplicity 2 at x=3, roots of multiplicity 1 at x=0 and x=-1, and passing through the point (5,48). Learn how to structure the polynomial and solve for the constant a.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation