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Chapter 3: Problem 60

Find a polynomial of the specified degree that has the given zeros. Degree \(5 ; \quad\) zeros \(-2,-1,0,1,2\)

Short Answer

Expert verified
The polynomial is \(P(x) = x^5 - 9x^3 + 4x\).

Step by step solution

01

Write Down the General Form of the Polynomial

The polynomial is expressed as a product of its factors, each corresponding to a zero. If a polynomial has zeros at \(-2, -1, 0, 1,\) and \(2\), its general form is \[ P(x) = a(x + 2)(x + 1)x(x - 1)(x - 2) \] where \(a\) is a non-zero constant.
02

Expand the Polynomial

First, multiply the factors in two pairs and then include the standalone term:\[ (x + 2)(x + 1) = x^2 + 3x + 2 \]\[ (x - 1)(x - 2) = x^2 - 3x + 2 \]Multiplying these results gives:\[ (x^2 + 3x + 2)(x^2 - 3x + 2) = x^4 - 9x^2 + 4 \]
03

Multiply by the Remaining Factor

Now, include the remaining factor \(x\) to the expression from Step 2:\[ x(x^4 - 9x^2 + 4) = x^5 - 9x^3 + 4x \]
04

Consider the Leading Coefficient

The constant \(a\) can be any non-zero number, typically chosen to be 1 for simplicity unless stated otherwise. Thus, the polynomial with the given zeros is:\[ P(x) = x^5 - 9x^3 + 4x \]

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Key Concepts

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

Degree of a Polynomial
Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication operations. One of the key characteristics of a polynomial is its degree.
  • The degree of a polynomial is the highest power of the variable within the expression.
  • For instance, in the polynomial expression \( x^5 - 9x^3 + 4x \), the degree is 5, because the term \( x^5 \) has the highest exponent.
  • The degree gives an idea of how the polynomial behaves as the variable gets larger or smaller.
Understanding the degree of a polynomial is crucial because it provides insight into the polynomial's shape and long-term behavior. In this specific exercise, we aim to construct a polynomial with a degree of 5.
Zeros of a Polynomial
The zeros (or roots) of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these zeros is fundamental in understanding the solutions and behavior of polynomial equations.
  • Zeros are typically obtained by solving the equation where the polynomial is equal to zero, \( P(x) = 0 \).
  • For example, in our exercise, the polynomial \( P(x) \) has the zeros \(-2, -1, 0, 1,\) and \(2\).
  • These zeros indicate where the polynomial will intersect the x-axis on a graph.
Each zero contributes a factor to the polynomial, such that if \( c \) is a zero, then \( x-c \) is a factor. In this exercise, these zeros help define the factors that form our polynomial.
Expanding Polynomials
Once we have a polynomial in its factored form, expanding it into a standard polynomial form involves multiplying the factors together.
  • Expanding is the process where each factor is multiplied with every other factor to simplify and combine them into a single polynomial expression.
  • This is done step-by-step, often starting with two simpler factors and then progressively including others.
  • In our solution, we first expanded \((x+2)(x+1)\) and then \((x-1)(x-2)\), and combined these into larger expressions.
  • Afterwards, the result is multiplied with the remaining linear factor, \(x\), to get a single polynomial expression.
This process is crucially important as it transforms the polynomial from an easily factorable form to one that is ready for analysis such as graphing or deriving further properties.
Factored Form of a Polynomial
A polynomial in its factored form is expressed as a product of its linear factors, each corresponding to a zero of the polynomial. This form is quite useful for a number of reasons.
  • Factored form directly shows the zeros, making it easy to understand the roots of the polynomial.
  • For example, given the zeros \(-2, -1, 0, 1,\) and \(2\), the factored form is \((x+2)(x+1)x(x-1)(x-2)\).
  • Handling polynomials in factored form is advantageous during division, simplification, and finding least common multiples.
The conversion of a polynomial into its factored form is a powerful tool in algebra, which aids in solving polynomial equations straightforwardly by setting each factor equal to zero.

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Most popular questions from this chapter

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$$

A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in hours since giving the drug), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$c(t)=\frac{5 t}{t^{2}+1}$$ Graph the function \(c\) with a graphing device. (a) What is the highest concentration of drug that is reached in the patient's bloodstream? (b) What happens to the drug concentration after a long period of time? (c) How long does it take for the concentration to drop below \(0.3 \mathrm{mg} / \mathrm{L} ?\)

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$

After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$c(t)=\frac{30 t}{t^{2}+2}$$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, g(x)=1-x^{2}$$
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