Algebra and Trigonometry: Structure and Method, Book 2
Algebra and Trigonometry: Structure and Method, Book 2
2000th Edition
ISBN: 9780395977255
Author: MCDOUGAL LITTEL
Publisher: McDougal Littell
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 8.6, Problem 19WE
To determine

To make the list of possibilities of nature of roots of equation.

Expert Solution & Answer
Check Mark

Answer to Problem 19WE

Since P(x)=0 has five roots in all, there are four possibilities.

    Number of Positive real roots Number of negative real roots Number of imaginary roots
    302
    320
    104
    122

Explanation of Solution

Given information:

The given equation is x5x3x2+x2=0

Formula used:

Descartes’ Rule of Signs

Let P(x) be a simplified polynomial with real coefficients and terms arranged in decreasing degree of x.

  1. The number of positive real roots of P(x)=0 equals to the number of variations of sign of P(x) or is fewer than this number by an even integer.
  2. The number of negative real roots of P(x)=0 equals to the number of variations of sign of P(x) or is fewer than this number by an even integer.

Calculation :

First write the polynomial

  P(x)=x5x3x2+x2

  1. P(x) Has only 3 variations in sign, so by part (1) of Descartes’ rule the number of positive real roots of P(x)=0 is 3 or 1.
  2. P(x)=x5+x3x2x2=0

      P(x) Has two variations in sign, so by part (2) of Descartes’ rule the number of negative real roots of P(x)=0 is 2 or 0.

  3. Since P(x)=0 has five roots in all, there are four possibilities.
    Number of Positive real roots Number of negative real roots Number of imaginary roots
    302
    320
    104
    122

Chapter 8 Solutions

Algebra and Trigonometry: Structure and Method, Book 2

Ch. 8.1 - Prob. 11OECh. 8.1 - Prob. 12OECh. 8.1 - Prob. 1WECh. 8.1 - Prob. 2WECh. 8.1 - Prob. 3WECh. 8.1 - Prob. 4WECh. 8.1 - Prob. 5WECh. 8.1 - Prob. 6WECh. 8.1 - Prob. 7WECh. 8.1 - Prob. 8WECh. 8.1 - Prob. 9WECh. 8.1 - Prob. 10WECh. 8.1 - Prob. 11WECh. 8.1 - Prob. 12WECh. 8.1 - Prob. 13WECh. 8.1 - Prob. 14WECh. 8.1 - Prob. 15WECh. 8.1 - Prob. 16WECh. 8.1 - Prob. 17WECh. 8.1 - Prob. 18WECh. 8.1 - Prob. 19WECh. 8.1 - Prob. 20WECh. 8.1 - Prob. 21WECh. 8.1 - Prob. 22WECh. 8.1 - Prob. 23WECh. 8.1 - Prob. 24WECh. 8.1 - Prob. 25WECh. 8.1 - Prob. 1PCh. 8.1 - Prob. 2PCh. 8.1 - Prob. 3PCh. 8.1 - Prob. 4PCh. 8.1 - Prob. 5PCh. 8.1 - Prob. 6PCh. 8.1 - Prob. 7PCh. 8.1 - Prob. 8PCh. 8.1 - Prob. 9PCh. 8.1 - Prob. 10PCh. 8.1 - Prob. 11PCh. 8.1 - Prob. 12PCh. 8.1 - Prob. 13PCh. 8.1 - Prob. 14PCh. 8.1 - Prob. 15PCh. 8.1 - Prob. 16PCh. 8.1 - Prob. 17PCh. 8.1 - Prob. 18PCh. 8.1 - Prob. 19PCh. 8.1 - Prob. 20PCh. 8.1 - Prob. 1MRECh. 8.1 - Prob. 2MRECh. 8.1 - Prob. 3MRECh. 8.1 - Prob. 4MRECh. 8.1 - Prob. 5MRECh. 8.1 - Prob. 6MRECh. 8.1 - Prob. 7MRECh. 8.1 - Prob. 8MRECh. 8.1 - Prob. 9MRECh. 8.2 - Prob. 1OECh. 8.2 - Prob. 2OECh. 8.2 - Prob. 3OECh. 8.2 - Prob. 4OECh. 8.2 - Prob. 5OECh. 8.2 - Prob. 6OECh. 8.2 - Prob. 7OECh. 8.2 - Prob. 8OECh. 8.2 - Prob. 9OECh. 8.2 - Prob. 10OECh. 8.2 - Prob. 11OECh. 8.2 - Prob. 12OECh. 8.2 - Prob. 13OECh. 8.2 - Prob. 14OECh. 8.2 - Prob. 1WECh. 8.2 - Prob. 2WECh. 8.2 - Prob. 3WECh. 8.2 - Prob. 4WECh. 8.2 - Prob. 5WECh. 8.2 - Prob. 6WECh. 8.2 - Prob. 7WECh. 8.2 - Prob. 8WECh. 8.2 - Prob. 9WECh. 8.2 - Prob. 10WECh. 8.2 - Prob. 1PCh. 8.2 - Prob. 2PCh. 8.2 - Prob. 3PCh. 8.2 - Prob. 4PCh. 8.2 - Prob. 5PCh. 8.2 - Prob. 6PCh. 8.2 - Prob. 7PCh. 8.2 - Prob. 8PCh. 8.2 - Prob. 9PCh. 8.2 - Prob. 10PCh. 8.2 - Prob. 11PCh. 8.2 - Prob. 12PCh. 8.2 - Prob. 13PCh. 8.2 - Prob. 14PCh. 8.2 - Prob. 1STCh. 8.2 - Prob. 2STCh. 8.2 - Prob. 3STCh. 8.2 - Prob. 4STCh. 8.2 - Prob. 5STCh. 8.2 - Prob. 6STCh. 8.2 - Prob. 7STCh. 8.3 - Prob. 1OECh. 8.3 - Prob. 2OECh. 8.3 - Prob. 3OECh. 8.3 - Prob. 4OECh. 8.3 - Prob. 5OECh. 8.3 - Prob. 6OECh. 8.3 - Prob. 7OECh. 8.3 - Prob. 8OECh. 8.3 - Prob. 9OECh. 8.3 - Prob. 10OECh. 8.3 - Prob. 11OECh. 8.3 - Prob. 12OECh. 8.3 - Prob. 1WECh. 8.3 - Prob. 2WECh. 8.3 - Prob. 3WECh. 8.3 - Prob. 4WECh. 8.3 - Prob. 5WECh. 8.3 - Prob. 6WECh. 8.3 - Prob. 7WECh. 8.3 - Prob. 8WECh. 8.3 - Prob. 9WECh. 8.3 - Prob. 10WECh. 8.3 - Prob. 11WECh. 8.3 - Prob. 12WECh. 8.3 - Prob. 13WECh. 8.3 - Prob. 14WECh. 8.3 - Prob. 15WECh. 8.3 - Prob. 16WECh. 8.3 - Prob. 17WECh. 8.3 - Prob. 18WECh. 8.3 - Prob. 19WECh. 8.3 - Prob. 20WECh. 8.3 - Prob. 21WECh. 8.3 - Prob. 22WECh. 8.3 - Prob. 23WECh. 8.3 - Prob. 24WECh. 8.3 - Prob. 25WECh. 8.3 - Prob. 26WECh. 8.3 - Prob. 27WECh. 8.3 - Prob. 28WECh. 8.3 - Prob. 29WECh. 8.3 - Prob. 30WECh. 8.3 - Prob. 31WECh. 8.3 - Prob. 32WECh. 8.3 - Prob. 33WECh. 8.3 - Prob. 34WECh. 8.3 - Prob. 1MRECh. 8.3 - Prob. 2MRECh. 8.3 - Prob. 3MRECh. 8.3 - Prob. 4MRECh. 8.3 - Prob. 5MRECh. 8.3 - Prob. 6MRECh. 8.3 - Prob. 7MRECh. 8.3 - Prob. 8MRECh. 8.4 - Prob. 1OECh. 8.4 - Prob. 2OECh. 8.4 - Prob. 3OECh. 8.4 - Prob. 4OECh. 8.4 - Prob. 5OECh. 8.4 - Prob. 6OECh. 8.4 - Prob. 1WECh. 8.4 - Prob. 2WECh. 8.4 - Prob. 3WECh. 8.4 - Prob. 4WECh. 8.4 - Prob. 5WECh. 8.4 - Prob. 6WECh. 8.4 - Prob. 7WECh. 8.4 - Prob. 8WECh. 8.4 - Prob. 9WECh. 8.4 - Prob. 10WECh. 8.4 - Prob. 11WECh. 8.4 - Prob. 12WECh. 8.4 - Prob. 13WECh. 8.4 - Prob. 14WECh. 8.4 - Prob. 15WECh. 8.4 - Prob. 16WECh. 8.4 - Prob. 17WECh. 8.4 - Prob. 18WECh. 8.4 - Prob. 19WECh. 8.4 - Prob. 20WECh. 8.4 - Prob. 21WECh. 8.4 - Prob. 22WECh. 8.4 - Prob. 23WECh. 8.4 - Prob. 24WECh. 8.4 - Prob. 1ECh. 8.4 - Prob. 2ECh. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.5 - Prob. 1OECh. 8.5 - Prob. 2OECh. 8.5 - Prob. 3OECh. 8.5 - Prob. 4OECh. 8.5 - Prob. 5OECh. 8.5 - Prob. 6OECh. 8.5 - Prob. 1WECh. 8.5 - Prob. 2WECh. 8.5 - Prob. 3WECh. 8.5 - Prob. 4WECh. 8.5 - Prob. 5WECh. 8.5 - Prob. 6WECh. 8.5 - Prob. 7WECh. 8.5 - Prob. 8WECh. 8.5 - Prob. 9WECh. 8.5 - Prob. 10WECh. 8.5 - Prob. 11WECh. 8.5 - Prob. 12WECh. 8.5 - Prob. 13WECh. 8.5 - Prob. 14WECh. 8.5 - Prob. 15WECh. 8.5 - Prob. 16WECh. 8.5 - Prob. 17WECh. 8.5 - Prob. 18WECh. 8.5 - Prob. 19WECh. 8.5 - Prob. 20WECh. 8.5 - Prob. 21WECh. 8.5 - Prob. 22WECh. 8.5 - Prob. 23WECh. 8.5 - Prob. 24WECh. 8.5 - Prob. 25WECh. 8.5 - Prob. 26WECh. 8.5 - Prob. 27WECh. 8.5 - Prob. 28WECh. 8.5 - Prob. 29WECh. 8.5 - Prob. 30WECh. 8.5 - Prob. 31WECh. 8.5 - Prob. 32WECh. 8.5 - Prob. 33WECh. 8.5 - Prob. 34WECh. 8.5 - Prob. 35WECh. 8.5 - Prob. 36WECh. 8.5 - Prob. 37WECh. 8.5 - Prob. 38WECh. 8.5 - Prob. 39WECh. 8.5 - Prob. 1MRECh. 8.5 - Prob. 2MRECh. 8.5 - Prob. 3MRECh. 8.5 - Prob. 4MRECh. 8.5 - Prob. 5MRECh. 8.5 - Prob. 6MRECh. 8.5 - Prob. 7MRECh. 8.5 - Prob. 8MRECh. 8.5 - Prob. 9MRECh. 8.6 - Prob. 1OECh. 8.6 - Prob. 2OECh. 8.6 - Prob. 3OECh. 8.6 - Prob. 4OECh. 8.6 - Prob. 5OECh. 8.6 - Prob. 6OECh. 8.6 - Prob. 7OECh. 8.6 - Prob. 8OECh. 8.6 - Prob. 9OECh. 8.6 - Prob. 10OECh. 8.6 - Prob. 11OECh. 8.6 - Prob. 12OECh. 8.6 - Prob. 13OECh. 8.6 - Prob. 14OECh. 8.6 - Prob. 15OECh. 8.6 - Prob. 1WECh. 8.6 - Prob. 2WECh. 8.6 - Prob. 3WECh. 8.6 - Prob. 4WECh. 8.6 - Prob. 5WECh. 8.6 - Prob. 6WECh. 8.6 - Prob. 7WECh. 8.6 - Prob. 8WECh. 8.6 - Prob. 9WECh. 8.6 - Prob. 10WECh. 8.6 - Prob. 11WECh. 8.6 - Prob. 12WECh. 8.6 - Prob. 13WECh. 8.6 - Prob. 14WECh. 8.6 - Prob. 15WECh. 8.6 - Prob. 16WECh. 8.6 - Prob. 17WECh. 8.6 - Prob. 18WECh. 8.6 - Prob. 19WECh. 8.6 - Prob. 20WECh. 8.6 - Prob. 21WECh. 8.6 - Prob. 22WECh. 8.6 - Prob. 23WECh. 8.6 - Prob. 24WECh. 8.6 - Prob. 25WECh. 8.6 - Prob. 26WECh. 8.6 - Prob. 27WECh. 8.6 - Prob. 28WECh. 8.6 - Prob. 29WECh. 8.6 - Prob. 1STCh. 8.6 - Prob. 2STCh. 8.6 - Prob. 3STCh. 8.6 - Prob. 4STCh. 8.6 - Prob. 5STCh. 8.6 - Prob. 6STCh. 8.6 - Prob. 7STCh. 8.7 - Prob. 1OECh. 8.7 - Prob. 2OECh. 8.7 - Prob. 3OECh. 8.7 - Prob. 4OECh. 8.7 - Prob. 5OECh. 8.7 - Prob. 6OECh. 8.7 - Prob. 7OECh. 8.7 - Prob. 8OECh. 8.7 - Prob. 9OECh. 8.7 - Prob. 10OECh. 8.7 - Prob. 11OECh. 8.7 - Prob. 12OECh. 8.7 - Prob. 13OECh. 8.7 - Prob. 14OECh. 8.7 - Prob. 15OECh. 8.7 - Prob. 16OECh. 8.7 - Prob. 1WECh. 8.7 - Prob. 2WECh. 8.7 - Prob. 3WECh. 8.7 - Prob. 4WECh. 8.7 - Prob. 5WECh. 8.7 - Prob. 6WECh. 8.7 - Prob. 7WECh. 8.7 - Prob. 8WECh. 8.7 - Prob. 9WECh. 8.7 - Prob. 10WECh. 8.7 - Prob. 11WECh. 8.7 - Prob. 12WECh. 8.7 - Prob. 13WECh. 8.7 - Prob. 14WECh. 8.7 - Prob. 15WECh. 8.7 - Prob. 16WECh. 8.7 - Prob. 17WECh. 8.7 - Prob. 18WECh. 8.7 - Prob. 19WECh. 8.7 - Prob. 20WECh. 8.7 - Prob. 21WECh. 8.7 - Prob. 22WECh. 8.7 - Prob. 23WECh. 8.7 - Prob. 24WECh. 8.7 - Prob. 25WECh. 8.7 - Prob. 26WECh. 8.7 - Prob. 27WECh. 8.7 - Prob. 28WECh. 8.7 - Prob. 1MRECh. 8.7 - Prob. 2MRECh. 8.7 - Prob. 3MRECh. 8.7 - Prob. 4MRECh. 8.7 - Prob. 5MRECh. 8.7 - Prob. 6MRECh. 8.7 - Prob. 7MRECh. 8.7 - Prob. 8MRECh. 8.7 - Prob. 9MRECh. 8.7 - Prob. 10MRECh. 8.8 - Prob. 1OECh. 8.8 - Prob. 2OECh. 8.8 - Prob. 3OECh. 8.8 - Prob. 4OECh. 8.8 - Prob. 5OECh. 8.8 - Prob. 6OECh. 8.8 - Prob. 7OECh. 8.8 - Prob. 8OECh. 8.8 - Prob. 1WECh. 8.8 - Prob. 2WECh. 8.8 - Prob. 3WECh. 8.8 - Prob. 4WECh. 8.8 - Prob. 5WECh. 8.8 - Prob. 6WECh. 8.8 - Prob. 7WECh. 8.8 - Prob. 8WECh. 8.8 - Prob. 9WECh. 8.8 - Prob. 10WECh. 8.8 - Prob. 11WECh. 8.8 - Prob. 12WECh. 8.8 - Prob. 13WECh. 8.8 - Prob. 14WECh. 8.8 - Prob. 15WECh. 8.8 - Prob. 16WECh. 8.8 - Prob. 17WECh. 8.8 - Prob. 18WECh. 8.8 - Prob. 19WECh. 8.8 - Prob. 20WECh. 8.8 - Prob. 21WECh. 8.8 - Prob. 22WECh. 8.8 - Prob. 23WECh. 8.8 - Prob. 24WECh. 8.8 - Prob. 25WECh. 8.8 - Prob. 26WECh. 8.8 - Prob. 27WECh. 8.8 - Prob. 28WECh. 8.8 - Prob. 29WECh. 8.8 - Prob. 30WECh. 8.8 - Prob. 1ECh. 8.8 - Prob. 2ECh. 8.8 - Prob. 3ECh. 8.8 - Prob. 4ECh. 8.9 - Prob. 1OECh. 8.9 - Prob. 2OECh. 8.9 - Prob. 3OECh. 8.9 - Prob. 1WECh. 8.9 - Prob. 2WECh. 8.9 - Prob. 3WECh. 8.9 - Prob. 4WECh. 8.9 - Prob. 5WECh. 8.9 - Prob. 6WECh. 8.9 - Prob. 7WECh. 8.9 - Prob. 8WECh. 8.9 - Prob. 9WECh. 8.9 - Prob. 10WECh. 8.9 - Prob. 11WECh. 8.9 - Prob. 12WECh. 8.9 - Prob. 13WECh. 8.9 - Prob. 14WECh. 8.9 - Prob. 15WECh. 8.9 - Prob. 16WECh. 8.9 - Prob. 17WECh. 8.9 - Prob. 18WECh. 8.9 - Prob. 19WECh. 8.9 - Prob. 20WECh. 8.9 - Prob. 21WECh. 8.9 - Prob. 22WECh. 8.9 - Prob. 23WECh. 8.9 - Prob. 24WECh. 8.9 - Prob. 25WECh. 8.9 - Prob. 26WECh. 8.9 - Prob. 27WECh. 8.9 - Prob. 28WECh. 8.9 - Prob. 29WECh. 8.9 - Prob. 30WECh. 8.9 - Prob. 1CECh. 8.9 - Prob. 2CECh. 8.9 - Prob. 3CECh. 8.9 - Prob. 4CECh. 8.9 - Prob. 5CECh. 8.9 - Prob. 1STCh. 8.9 - Prob. 2STCh. 8.9 - Prob. 3STCh. 8.9 - Prob. 4STCh. 8.9 - Prob. 5STCh. 8 - Prob. 1CRCh. 8 - Prob. 2CRCh. 8 - Prob. 3CRCh. 8 - Prob. 4CRCh. 8 - Prob. 5CRCh. 8 - Prob. 6CRCh. 8 - Prob. 7CRCh. 8 - Prob. 8CRCh. 8 - Prob. 9CRCh. 8 - Prob. 10CRCh. 8 - Prob. 11CRCh. 8 - Prob. 12CRCh. 8 - Prob. 13CRCh. 8 - Prob. 1CTCh. 8 - Prob. 2CTCh. 8 - Prob. 3CTCh. 8 - Prob. 4CTCh. 8 - Prob. 5CTCh. 8 - Prob. 6CTCh. 8 - Prob. 7CTCh. 8 - Prob. 8CTCh. 8 - Prob. 9CT
Knowledge Booster
Background pattern image
Algebra
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:PEARSON
Text book image
Contemporary Abstract Algebra
Algebra
ISBN:9781305657960
Author:Joseph Gallian
Publisher:Cengage Learning
Text book image
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Text book image
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:9780135163078
Author:Michael Sullivan
Publisher:PEARSON
Text book image
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:9780980232776
Author:Gilbert Strang
Publisher:Wellesley-Cambridge Press
Text book image
College Algebra (Collegiate Math)
Algebra
ISBN:9780077836344
Author:Julie Miller, Donna Gerken
Publisher:McGraw-Hill Education
10 - Roots of polynomials; Author: Technion;https://www.youtube.com/watch?v=88YUeigknNg;License: Standard YouTube License, CC-BY