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Chapter 3: Problem 40

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(3 ;\) zeros \(-6,0,\) and \(3,\) each of multiplicity 1

Short Answer

Expert verified
The polynomial function is given by \(f(x) = (x+6)x(x-3)\).

Step by step solution

01

Formulate the Zeros

First, we identify the zeros provided. In the case, the given zeros are -6, 0, and 3. Considering the multiplicity of each zero (which is 1 for all), we can formulate the corresponding factors as \( (x+6), (x-0),\) and \( (x-3) \) respectively.
02

Create the Polynomial

The polynomial is formed by multiplying these factors. The polynomial function is given by \(f(x) = a (x+6)(x-0)(x-3)\) where \(a\) is a constant.
03

Identify the Constant

In this situation, since the degree of the polynomial is 3, the leading coefficient 'a' is assumed to be 1 because the highest power x will have is equal to the degree of the polynomial. Therefore we have: \(f(x) = 1 * (x+6)(x)(x-3)\).

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Key Concepts

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

Degree of Polynomial
The degree of a polynomial is a fundamental characteristic as it reveals the polynomial's highest power of the variable, often referred to as the leading term. In simpler terms, it tells us how many times the largest variable appears multiplied by itself. When describing a polynomial's degree, we focus on:
  • The degree is determined by the term with the highest exponent.
  • A polynomial of degree 3 means the variable, typically \(x\), reaches at most \(x^3\).
  • The degree of 3 in the exercise example signifies that the polynomial will include a term \(x^3\).
This degree also tells us the number of solutions or roots the polynomial might have. For instance, a cubic (third-degree) polynomial will usually have three roots, which can be real or complex. These roots correspond directly to the zeros identified in the problem.
Multiplicity of Zeros
The multiplicity of zeros is a concept that essentially tells us how many times a particular solution, or zero, appears for a polynomial.
  • A zero with a multiplicity of 1 indicates a root that occurs once, generating a simple crossing of the \(x\)-axis at that point.
  • Higher multiplicities might produce a more intricate interaction with the \(x\)-axis, such as touching or oscillating behaviors.
  • In our exercise, each zero, -6, 0, and 3, has a multiplicity of 1.
Thus, every zero contributes one factor to the polynomial. For zeros \(x = -6\), \(x = 0\), and \(x = 3\), the factors will be \((x+6), (x),\) and \((x-3)\). When counting multiplicity, you add these factors once since they each have a multiplicity of one.
Factoring Polynomials
Factoring a polynomial involves breaking it down into a product of simpler polynomials, which usually represent its roots or zeros. This process is essential because it displays the polynomial in a form that clearly shows its roots. Let's delve into the importance:
  • Factoring is the reverse of expanding a polynomial. It helps to identify the roots easily by setting each factor to zero.
  • The roots derived from factoring correspond to the zeros where the polynomial intersects the \(x\)-axis. These are important for solving equations and understanding polynomial graphs.
  • Our polynomial from the exercise, through the identified zeros, is \(f(x) = (x+6)(x)(x-3)\). This form is reached by factoring the cubic polynomial.
By factoring, you gain insight into the polynomial's structure, making it easier to graph and analyze. This process of coming up with factors acts like a detective story revealing the polynomial's nature and solutions.

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Most popular questions from this chapter

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=2 x^{5}-6 x^{3}+7 x^{2}-8$$

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