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)=x4+6x3−18x2; between 2 and 3
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 36
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
First, write down the inequality clearly: \(x^3 + 2x^2 - 4x - 8 \geq 0\).
Next, factor the cubic polynomial on the left side. Start by looking for rational roots using the Rational Root Theorem, which suggests possible roots among the factors of the constant term (±1, ±2, ±4, ±8).
Once a root is found, use polynomial division or synthetic division to factor the cubic polynomial into a product of a linear factor and a quadratic factor.
After factoring completely, set each factor greater than or equal to zero to find critical points and determine intervals to test for the inequality.
Test values from each interval in the original inequality to determine where the expression is greater than or equal to zero, then express the solution set in interval notation and graph it on the real number line.
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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 symbols (>, <, ≥, ≤). 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.
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Linear Inequalities
Factoring Polynomials
Factoring is the process of breaking down a polynomial into simpler polynomials (factors) that multiply to give the original. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.
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07:30Introduction to Factoring Polynomials
Interval Notation and Number Line Graphing
Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial satisfies the inequality, using open or closed dots to indicate whether endpoints are included.
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05:18Interval Notation
Related Practice
Textbook Question
Use the Rational Zero Theorem to list all possible rational zeros for each given function.
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Textbook Question
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.f(x)=2x4−4x2+1; between -1 and 0
Textbook Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)
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Textbook Question
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x3−7x2−2x+5;f(−3)
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Textbook Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function.