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Chapter 3: Problem 43

Create a polynomial \(p\) which has the desired characteristics. You may leave the polynomial in factored form. \(\bullet\) The solutions to \(p(x)=0\) are \(x=\pm 3\) and \(x=6\) \(\bullet\) The leading term of \(p(x)\) is \(7 x^{4}\) \(\bullet\) The point (-3,0) is a local minimum on the graph of \(y=p(x)\).

Short Answer

Expert verified
The polynomial is \( p(x) = 7(x + 3)^2(x - 3)(x - 6) \).

Step by step solution

01

Identify the Solutions and Their Multiplicities

Given that the solutions to \(p(x)=0\) are \(x=\pm 3\) and \(x=6\), we can express \(p(x)\) in terms of its factors: \((x + 3)\), \((x - 3)\), and \((x - 6)\). Since \((-3,0)\) is a local minimum, this implies that the factor \((x + 3)\) must have an even multiplicity. A simple option is to use a multiplicity of 2 for \((x + 3)\).
02

Ensure Correct Leading Term

The desired leading term is \(7x^4\). Given the factors \((x + 3)^2\), \((x - 3)\), and \((x - 6)\), we have the polynomial \((x + 3)^2(x - 3)(x - 6)\), which would originally expand into a leading term of \(x^4\). To achieve the leading term of \(7x^4\), we multiply the entire expression by 7: \[ p(x) = 7(x + 3)^2(x - 3)(x - 6). \]
03

Translate Polynomial to Factored Form

The polynomial \(p(x) = 7(x + 3)^2(x - 3)(x - 6)\) meets all given conditions. It remains in the required factored form, with each factor represented as determined from the roots. The factor \((x+3)^2\) ensures that the double root condition satisfies the local minimum at \(x = -3\).

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Key Concepts

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

Roots and Zeros
Roots, sometimes referred to as zeros, are the values of a polynomial that make the polynomial equal to zero. In simple terms, these are the values of \(x\) for which the equation \(p(x) = 0\) holds true.

For example, given the problem, the roots are \(x = -3\), \(x = 3\), and \(x = 6\). This means that if you plug these values into the polynomial, the result will be zero. Roots can represent points where a graph intersects the x-axis. These are crucial for finding the shape and position of the graph of a polynomial function.
  • For \(x = -3\): The factor would be \((x + 3)\).
  • For \(x = 3\): The factor would be \((x - 3)\).
  • For \(x = 6\): The factor would be \((x - 6)\).
Knowing this helps you construct the fundamental framework of the polynomial by forming these factors.
Leading Term
The leading term is the term in a polynomial with the highest degree, which is crucial as it determines the end behavior of the polynomial's graph. Here, the leading term specified is \(7x^4\).

This insight tells us that the polynomial is of degree 4, which suggests certain characteristics about its graph, such as the types of turning points it may have.
  • The factor \((x+3)^2(x-3)(x-6)\) originally gives a leading term of \(x^4\).
  • To match the desired leading term, you multiply the entire polynomial by 7, resulting in \(7(x+3)^2(x-3)(x-6)\), providing a leading term of \(7x^4\).
Essentially, this adjustment ensures the graph behaves in specific prescribed ways, particularly as x approaches infinity or negative infinity.
Local Minimum
A local minimum refers to a point on a graph of a function where the function value is lower than those surrounding it, but not necessarily the lowest overall. In the given polynomial, the point \((-3,0)\) represents a local minimum.

This means that around \(x = -3\), the graph dips to a point that is lower than points in the immediate vicinity, creating a valley-like appearance. Ensuring a local minimum at a root requires that the corresponding factor have an even multiplicity.

In this case, \((x + 3)\) must appear twice in the polynomial as \((x + 3)^2\). The even multiplicity causes the graph to "touch" the x-axis at \((-3, 0)\) and rise again, instead of crossing it.
Multiplicity of Roots
The multiplicity of a root refers to how many times a particular root is repeated in the factorization of a polynomial. It affects the graph's behavior at the root. With a multiplicity of 1, the graph will cross the x-axis at the root. With higher multiplicities, the root may just touch the axes or form an inflection point.
  • For \(x = 3\) and \(x = 6\): The multiplicity is 1, meaning the graph will cross the x-axis at these points.
  • For \(x = -3\): The multiplicity is 2, which means here the graph just "touches" the x-axis and creates a local minimum, without crossing it.
Higher multiplicities signify repeated factors, which in turn affect how sharply or softly the curve approaches and interacts with the x-axis at these points.

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Most popular questions from this chapter

Use polynomial long division to perform the indicated division. Write the polynomial in the form \(p(x)=d(x) q(x)+r(x)\). \(\left(4 x^{2}-x-23\right) \div\left(x^{2}-1\right)\)

You are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(x^{3}-24 x^{2}+192 x-512, \quad c=8\)

Use synthetic division to perform the indicated division. Write the polynomial in the form \(p(x)=d(x) q(x)+r(x)\). \(\left(x^{6}-6 x^{4}+12 x^{2}-8\right) \div(x+\sqrt{2})\)

You are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(2 x^{3}-x^{2}-10 x+5, \quad c=\frac{1}{2}\)

Find the real zeros of the polynomial using the techniques specified by your instructor. State the multiplicity of each real zero. \(f(x)=6 x^{4}-5 x^{3}-9 x^{2}\)
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