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Chapter 3: Problem 25

Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$Q(x)=x^{4}+2 x^{2}+1$$

Short Answer

Expert verified
Zeros are \( x = i \) and \( x = -i \) with multiplicity 2.

Step by step solution

01

Recognize the polynomial form

The given polynomial is \( Q(x) = x^4 + 2x^2 + 1 \). Notice it has the form of a quadratic equation in terms of \(x^2\). You can rewrite it as \((x^2)^2 + 2(x^2) + 1\).
02

Try substitution for ease

Let \( u = x^2 \). The polynomial then becomes \( u^2 + 2u + 1 \). This form makes it easier to apply quadratic methods.
03

Factor the quadratic expression

Factor the quadratic \( u^2 + 2u + 1 \). This can be rewritten as \((u+1)^2\) because it fits the formula \( a^2 + 2ab + b^2 = (a+b)^2 \), where \( a=u \) and \( b=1 \).
04

Resubstitute back to x

Since \( u = x^2 \), replace \( u \) in the factorization \((u+1)^2\) with \( x^2 \) to get \((x^2 + 1)^2\).
05

Solve for zeros by setting factors to zero

Set each factor equal to zero: \((x^2 + 1) = 0\). This leads to \( x^2 = -1 \). The solutions to this equation are \( x = i \) and \( x = -i \), where \( i \) is the imaginary unit.
06

Determine the multiplicity of the zeros

Since the factor \((x^2 + 1)\) appears squared, each of the zeros \( x = i \) and \( x = -i \) have a multiplicity of 2.

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Key Concepts

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

Quadratic Substitution
Quadratic substitution is a technique used to simplify polynomial expressions, especially when dealing with higher degree polynomials that display a clear quadratic form nested within them. In our exercise, the polynomial \( Q(x) = x^4 + 2x^2 + 1 \) is transformed using this method.

Here's how it works:
  • Identify patterns that resemble a quadratic equation. In this case, \( (x^2)^2 + 2(x^2) + 1 \) resembles the standard quadratic form \( a^2 + 2ab + b^2 \).
  • Introduce a substitution variable. We let \( u = x^2 \), giving us a simpler quadratic \( u^2 + 2u + 1 \).
  • Once transformed, solve or factor the equation as a standard quadratic.
  • After solving, substitute back the original variable to complete the solution process.
Quadratic substitution transforms a complex polynomial into a manageable one, allowing for straightforward factorization steps.
Zeros of a Polynomial
The zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. In other words, they are the roots or solutions to the polynomial equation. In solving our problem, the polynomial \( Q(x) = (x^2 + 1)^2 \) leads us to the equation \( (x^2 + 1) = 0 \).

Solving for zeros involves:
  • Setting each factor of the polynomial equal to zero, i.e., \( x^2 + 1 = 0 \).
  • Recognizing any complex solutions. Here, we get \( x^2 = -1 \), which gives solutions \( x = i \) and \( x = -i \), using \( i \) as the imaginary unit.
  • In some cases, when factors have real roots, additional steps may be required to find real values of \( x \).
By finding the zeros, one can understand the behavior and intersection points of a polynomial on a graph.
Multiplicity of Zeros
Multiplicity refers to the number of times a particular zero appears in a polynomial function. In mathematical terms, if a zero of a polynomial corresponds to the factor \((x - a)^n\), then that zero "a" has a multiplicity of \( n \).

Let's apply this concept:
  • In our exercise, after factoring, we have \((x^2 + 1)^2\), indicating that the zeros \( x = i \) and \( x = -i \) each appear twice, hence a multiplicity of 2.
  • Zeros with a higher multiplicity affect the shape of the polynomial's graph. For instance, zeros with even multiplicity typically touch the x-axis and turn around, whereas zeros with odd multiplicity will cross the axis.
  • Identifying multiplicity can give insights into the symmetry and extremity in the polynomial's behavior.
Understanding multiplicity is key to interpreting the polynomial's graph and predicting its behavior across the x-axis.

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Most popular questions from this chapter

In this chapter we adopted the convention that in rational functions, the numerator and denominator don鈥檛 share a common factor. In this exercise we consider the graph of a rational function that doesn鈥檛 satisfy this rule. (a) Show that the graph of $$r(x)=\frac{3 x^{2}-3 x-6}{x-2}$$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint: Factor. What is the domain of \(r ?]\) (b) Graph the rational functions: $$\begin{aligned} &s(x)=\frac{x^{2}+x-20}{x+5}\\\ &t(x)=\frac{2 x^{2}-x-1}{x-1}\\\ &u(x)=\frac{x-2}{x^{2}-2 x} \end{aligned}$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{2 x^{3}+2 x}{x^{2}-1}$$

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right)$$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\). Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{4 x^{2}}{x^{2}-2 x-3}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}$$
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