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Chapter 5: Problem 92

Make up a polynomial function that has the following characteristics: crosses the \(x\) -axis at -1 and \(4,\) touches the \(x\) -axis at 0 and \(2,\) and is above the \(x\) -axis between 0 and \(2 .\) Give your polynomial to a fellow classmate and ask for a written critique.

Short Answer

Expert verified
A polynomial solution is \( P(x) = (x+1)(x-4)x^2(x-2)^2 \).

Step by step solution

01

- Determine the Zeroes

Identify the points where the polynomial crosses or touches the x-axis. The given points are -1, 0, 2, and 4. Crosses at -1 and 4, and touches at 0 and 2.
02

- Assign Multiplicities to Zeroes

Since the polynomial crosses the x-axis at -1 and 4, these zeroes have an odd multiplicity (1). The zeroes at 0 and 2 touch the x-axis without crossing, so these zeroes have an even multiplicity (2).
03

- Construct the Polynomial

Using the zeroes and their respective multiplicities, the polynomial can be written as:\[ P(x) = k(x+1)(x-4)(x^2)(x-2)^2 \]where k is a scaling factor. To satisfy the condition that the polynomial is above the x-axis between 0 and 2, choose a positive k.
04

- Simplify the Polynomial

Simplify the polynomial expression if desired. For example, let k=1:\[ P(x) = (x+1)(x-4)x^2(x-2)^2 \]Expanding this would give the simplified polynomial.
05

- Verify the Behavior

Confirm the polynomial meets the given criteria by checking the behavior at the specified points. Ensure the polynomial crosses or touches the x-axis correctly and is above the x-axis between 0 and 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zeroes
In polynomial functions, 'zeroes' (or roots) are the values of \( x \) where the polynomial equals zero. For the given exercise, we need to identify the points where the polynomial crosses or touches the x-axis. These points are the zeroes of the polynomial.
In this exercise, the zeroes provided are -1, 0, 2, and 4. Zeroes play an essential role in shaping the graph of the polynomial function.
Let's represent them:
  • The polynomial crosses the x-axis at -1 and 4. This means that the polynomial changes sign at these points.
  • The polynomial touches the x-axis at 0 and 2 without crossing it. This implies the value of the polynomial doesn't change signs at these points.
Understanding where these zeroes occur helps in constructing and analyzing the polynomial function accurately.
Multiplicities
Multiplicity refers to the number of times a particular root occurs in a polynomial. It significantly affects the shape and behavior of the graph.
For our polynomial:
  • A root with odd multiplicity (e.g., 1, 3) means the graph crosses the x-axis.
  • A root with even multiplicity (e.g., 2, 4) means the graph touches the x-axis but doesn't cross it.
In the given problem:
  • Zeroes at -1 and 4 have odd multiplicities (1) because the graph crosses the x-axis at these points.
  • Zeroes at 0 and 2 have even multiplicities (2) because the graph touches the x-axis at these points without crossing.
Recognizing and assigning the correct multiplicities help in plotting a more accurate graph in line with the problem's requirements.
Polynomial Construction
To construct a polynomial, we use the identified zeroes and their respective multiplicities. Here's the step-by-step process:
We start by writing the polynomial in its factored form:
  • The zero at x = -1, with an odd multiplicity, is written as \( (x + 1) \).
  • The zero at x = 4, with an odd multiplicity, is written as \( (x - 4) \).
  • The zero at x = 0, with an even multiplicity, is written as \( x^2 \).
  • The zero at x = 2, with an even multiplicity, is written as \( (x - 2)^2 \).
Combining these, the polynomial is constructed as: \[ P(x) = k (x + 1)(x - 4)x^2(x - 2)^2 \]Here, \( k \) is a scaling factor ensuring the polynomial fits the specific requirements of the problem. For simplicity, we can let \( k = 1 \) unless additional constraints are provided.
To meet the condition that the polynomial is above the x-axis between 0 and 2, \( k \) is chosen to be positive.
Polynomial Behavior
Analyzing the behavior of the polynomial involves verifying that it satisfies all given conditions:
  • At \( x = -1 \): The polynomial crosses the x-axis.
  • At \( x = 0 \): The polynomial touches the x-axis and does not cross it.
  • At \( x = 2 \): The polynomial touches the x-axis and does not cross it.
  • At \( x = 4 \): The polynomial crosses the x-axis.
Additionally, between \( x = 0 \) and \( x = 2 \), the polynomial is above the x-axis.
By plugging these points into our polynomial \[ P(x) = (x + 1)(x - 4)x^2(x - 2)^2 \], we can confirm this:
  • At \( x = -1, P(-1) = 0 \), the polynomial crosses the x-axis.
  • At \( x = 0, P(0) = 0 \), the polynomial touches the x-axis.
  • At \( x = 2, P(2) = 0 \), the polynomial touches the x-axis.
  • At \( x = 4, P(4) = 0 \), the polynomial crosses the x-axis.
Between these points, especially from 0 to 2, \[ P(x) \] is positive, confirming it is above the x-axis.
By carefully constructing and verifying the polynomial, we ensure it adheres to all given conditions.

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