The polynomial of degree 4, P(x) has a root of multiplicity 2 at a = 2 and roots of multiplicity 1 at a = 0 and a = -2. It goes through the point (5, 157.5). Find a formula for P(x). P(x) =

Calculus: Early Transcendentals
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:**

The polynomial of degree 4, \( P(x) \), has a root of multiplicity 2 at \( x = 2 \) and roots of multiplicity 1 at \( x = 0 \) and \( x = -2 \). It goes through the point \( (5, 157.5) \).

Find a formula for \( P(x) \).

**Task:**

\[ P(x) = \, \]

*Note: The problem requires finding the polynomial based on the given roots and point. The polynomial can be expressed as a product of linear factors raised to the power of their multiplicities. Substitute the given point to find any unknown coefficient.*
Transcribed Image Text:**Problem Statement:** The polynomial of degree 4, \( P(x) \), has a root of multiplicity 2 at \( x = 2 \) and roots of multiplicity 1 at \( x = 0 \) and \( x = -2 \). It goes through the point \( (5, 157.5) \). Find a formula for \( P(x) \). **Task:** \[ P(x) = \, \] *Note: The problem requires finding the polynomial based on the given roots and point. The polynomial can be expressed as a product of linear factors raised to the power of their multiplicities. Substitute the given point to find any unknown coefficient.*
Let \( f(x) = 1(x - 4)(x - 5)^2 \).

Fill out the table below. In the third column, type the word crosses for a zero where the curve crosses through to the other side of the \( x \)-axis, or touches for a zero where the curve approaches the \( x \)-axis, touches it, and reverses direction without crossing over it.

| zero | multiplicity | crosses or touches the \( x \)-axis |
|------|--------------|-------------------------------------|
|      | 1            |                                     |
| 5    | 2            |                                     |
Transcribed Image Text:Let \( f(x) = 1(x - 4)(x - 5)^2 \). Fill out the table below. In the third column, type the word crosses for a zero where the curve crosses through to the other side of the \( x \)-axis, or touches for a zero where the curve approaches the \( x \)-axis, touches it, and reverses direction without crossing over it. | zero | multiplicity | crosses or touches the \( x \)-axis | |------|--------------|-------------------------------------| | | 1 | | | 5 | 2 | |
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