Textbook Question
In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x5−x3−1; between 1 and 2
519
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 37
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
Verified step by step guidance1
Start by rewriting the inequality: \(x^3 - 3x^2 - 9x + 27 < 0\).
Factor the cubic polynomial if possible. Look for rational roots using the Rational Root Theorem and test possible roots such as \(\pm1\), \(\pm3\), \(\pm9\), \(\pm27\).
Once a root is found, use polynomial division or synthetic division to factor the cubic into a product of a linear factor and a quadratic factor.
Solve the quadratic factor by factoring further or using the quadratic formula to find all critical points (roots) where the expression equals zero.
Use the critical points to divide the real number line into intervals, then test a point from each interval in the original inequality to determine where the polynomial is less than zero. Express the solution set in interval notation.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.
Recommended video:
06:07
Linear Inequalities
Factoring Polynomials
Factoring is the process of breaking down a polynomial into simpler polynomials (factors) that multiply to give the original. Factoring helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign.
Recommended video:
Guided course
07:30Introduction to Factoring Polynomials
Interval Notation and Sign Analysis
Interval notation expresses solution sets as ranges of values on the number line. After finding the roots, sign analysis tests the polynomial's sign in intervals between roots to determine where the inequality holds, allowing the solution to be written accurately in interval notation.
Recommended video:
05:18Interval Notation
Related Practice
Textbook Question
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x4+5x3+5x2−5x−6;f(3)
517
views
Textbook Question
In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2−2
1007
views
Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
507
views
Textbook Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)
1051
views
Textbook Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function.
568
views