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Chapter 12: Problem 18

You are given a polynomial equation \(f(x)=0 .\) Specify the multiplicity of each repeated root. Then use a graphing utility to visually verify that the graph of \(y=f(x)\) is tangent to the \(x\) -axis at each repeated root. (a) \(x(x+1)^{2}(x-1)=0\) (b) \(x^{2}(x+1)^{2}(x-1)=0\) (c) \(x^{3}(x+1)^{2}(x-1)=0\)

Short Answer

Expert verified
(a) Multiplicities: 1, 2, 1; (b) 2, 2, 1; (c) 3, 2, 1. Verify with graph.

Step by step solution

01

Identify the Roots

For the given polynomial equation, identify the roots from the factored form. For \( (a) \) \( x(x+1)^{2}(x-1)=0 \), the roots are \( x=0, x=-1, \) and \( x=1 \). For \( (b) \) \( x^{2}(x+1)^{2}(x-1)=0 \), the roots remain the same: \( x=0, x=-1, \) and \( x=1 \). Similarly, for \( (c) \) \( x^{3}(x+1)^{2}(x-1)=0 \), the roots are, again, \( x=0, x=-1, \) and \( x=1 \).
02

Determine Multiplicities

For each equation, determine the multiplicity of each root by analyzing the exponents in the factored form. In \( (a) \), \( x=0 \) has multiplicity 1, \( x=-1 \) has multiplicity 2, and \( x=1 \) has multiplicity 1. In \( (b) \), \( x=0 \) has multiplicity 2, \( x=-1 \) has multiplicity 2, \( x=1 \) has multiplicity 1. In \( (c) \), \( x=0 \) has multiplicity 3, \( x=-1 \) has multiplicity 2, \( x=1 \) has multiplicity 1.
03

Understand Root Behavior

The behavior of the graph at each root depends on its multiplicity. For roots with odd multiplicity, the graph crosses the x-axis. For roots with even multiplicity, the graph is tangent to the x-axis.
04

Verify with Graphing

Use a graphing utility to plot the graphs of \( y = x(x+1)^{2}(x-1) \), \( y = x^{2}(x+1)^{2}(x-1) \), and \( y = x^{3}(x+1)^{2}(x-1) \). Verify that the even multiplicity roots \( x = -1 \) and in \( (b) \) \( x=0 \), and in \( (c) \) \( x=0 \) correspond to points where the graph is tangent to the x-axis. The other roots where the multiplicity is odd will cross the x-axis.

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Key Concepts

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

Roots of Equations
Identifying the roots of polynomial equations is a fundamental concept in algebra. Roots of an equation are values of the variable that make the entire equation equal to zero. In simpler terms, they are the points where the graph of the equation touches or crosses the x-axis. Consider the polynomial equation in factored form, like \(x(x+1)^{2}(x-1)=0\). Here, we can quickly identify the roots by looking at each factor. Each factor set equal to zero gives us a root:
  • \(x=0\) from the factor \(x\)
  • \(x=-1\) from the factor \((x+1)^{2}\)
  • \(x=1\) from the factor \((x-1)\)
These roots are critical in graphing and provide insights into the behavior of polynomials.
Multiplicity of Roots
The multiplicity of a root refers to the number of times a particular root appears for a given polynomial. In the factored form of the equation, the exponent on each factor shows the multiplicity of the corresponding root. A root with higher multiplicity affects the shape and behavior of the polynomial graph.

Take, for example, the polynomial \(x(x+1)^{2}(x-1)=0\). Here:
  • \(x=0\) has a multiplicity of 1 because it shows up as a single factor.
  • \(x=-1\) has a multiplicity of 2, indicating the graph will touch but not cross the x-axis at this point.
  • \(x=1\) has a multiplicity of 1.
Roots with even multiplicity, like \(x=-1\), imply that the graph is tangent to the x-axis, while odd multiplicities cause the graph to cross the x-axis.
Factored Form
The factored form of a polynomial equation can be immensely helpful for identifying roots and their multiplicities efficiently. Factoring means expressing the polynomial as a product of its linear factors. This format not only simplifies the solving process for the roots but also allows for easy interpretation of the polynomial's behavior.

Let's consider \(x^2(x+1)^2(x-1)=0\):
  • The factor \(x^2\) reveals the root \(x=0\) with a multiplicity of 2.
  • The factor \((x+1)^2\) gives the root \(x=-1\) with multiplicity 2.
  • The factor \((x-1)\) shows the root \(x=1\) with multiplicity 1.
By converting a polynomial to its factored form, one gains instant access to its roots and the insight into graphing its behavior effectively.

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Most popular questions from this chapter

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{4 x-5}{x^{4}+2 x^{3}+x^{2}-1}$$

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{2 x^{7}+3 x^{6}-2 x^{4}-4 x^{3}+2}{x^{4}+2 x^{3}+x^{2}-1}$$

Determine a value for \(a\) such that one root of the equation \(a x^{2}+x-1=0\) is five times the other.

Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccol貌 Tartaglia of Brescia gave Niccol貌 occasion to discover it. He / Tartaglial gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in /various / forms. - Girolamo Cardano, Ars Magna (Nuremberg, 1545 ) The quotation is from the translation of Ars Magna by T. Richard Witmer (New York: Dover Publications, 1993 ). In his book Ars Magna (The Great Art) the Renaissance mathematician Girolamo Cardano \((1501-1576)\) gave the following formula for a root of the equation \(x^{3}+a x=b\). $$x=\sqrt[3]{\frac{b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}-\sqrt[3]{\frac{-b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}$$ (a) Use this formula and your calculator to compute a root of the cubic equation \(x^{3}+3 x=76\) (b) Use a graph to check the answer in part (a). That is, graph the function \(y=x^{3}+3 x-76,\) and note the \(x-\) intercept. Also check the answer simply by substituting it in the equation \(x^{3}+3 x=76\)

This exercise outlines a proof of the rational roots theorem. At one point in the proof, we will need to rely on the following fact, which is proved in courses on number theory. FACT FROM NUMBER THEORY Suppose that \(A, B\), and \(C\) are integers and that \(A\) is a factor of the number \(B C .\) If \(A\) has no factor in common with \(C\) (other than 卤1 ), then \(A\) must be a factor of \(B\) (a) Let \(A=2, B=8,\) and \(C=5 .\) Verify that the fact from number theory is correct here. (b) Let \(A=20, B=8,\) and \(C=5 .\) Note that \(A\) is a factor of \(B C,\) but \(A\) is not a factor of \(B .\) Why doesn't this contradict the fact from number theory? (c) Now we're ready to prove the rational roots theorem. We begin with a polynomial equation with integer coefficients: \(-a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}=0 \quad\left(n \geq 1, a_{n} \neq 0\right)\) We assume that the rational number \(p / q\) is a root of the equation and that \(p\) and \(q\) have no common factors other than \(1 .\) Why is the following equation now true? \(a_{n}\left(\frac{p}{q}\right)^{n}+a_{n-1}\left(\frac{p}{q}\right)^{n-1}+\cdots+a_{1}\left(\frac{p}{q}\right) a_{0}=0\) (d) Show that the last equation in part (c) can be written \(p\left(a_{n} p^{n-1}+a_{n-1} q p^{n-2}+\cdots+a_{1} q^{n-1}\right)=-a_{0} q^{n}\) since \(p\) is a factor of the left-hand side of this last equation, \(p\) must also be a factor of the right-hand side. That is, \(p\) must be a factor of \(a_{0} q^{n}\). But since \(p\) and \(q\) have no common factors, neither do \(p\) and \(q^{n}\) Our fact from number theory now tells us that \(p\) must be a factor of \(a_{0}\), as we wished to show. (The proof that \(q\) is a factor of \(a_{n}\) is carried out in a similar manner.)
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