Textbook Question
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
2x3 + 11x2 < 7x + 6
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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 71
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
Verified step by step guidance1
Rewrite the given inequality \(2x^3 + 11x^2 \geq 7x + 6\) by bringing all terms to one side to set the inequality to zero: \(2x^3 + 11x^2 - 7x - 6 \geq 0\).
Recognize that the expression on the left side is the polynomial function \(f(x) = 2x^3 + 11x^2 - 7x - 6\) whose graph is provided.
Identify the x-values where \(f(x) = 0\) by looking at the points where the graph crosses the x-axis. These are the roots of the polynomial and critical points for the inequality.
Determine the intervals on the x-axis where the graph of \(f(x)\) is above or on the x-axis (i.e., where \(f(x) \geq 0\)). These intervals satisfy the inequality.
Express the solution set as the union of intervals where \(f(x) \geq 0\), based on the x-intercepts and the behavior of the graph between these points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions and Their Graphs
A polynomial function is an expression involving variables raised to whole-number exponents and their coefficients. The graph of a polynomial function shows its behavior, including where it crosses the x-axis (roots) and its general shape. Understanding the graph helps in visualizing solutions to inequalities involving the polynomial.
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Solving Polynomial Inequalities Using Graphs
To solve inequalities like f(x) ≥ 0 using a graph, identify where the graph lies above or on the x-axis. The x-values corresponding to these regions satisfy the inequality. The points where the graph touches or crosses the x-axis are critical points that divide the number line into intervals to test.
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Finding Roots of Polynomial Functions
Roots or zeros of a polynomial are the x-values where the function equals zero, i.e., where the graph crosses the x-axis. These roots are essential for solving inequalities because they mark boundaries between positive and negative values of the function. They can be found algebraically or estimated from the graph.
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Related Practice
Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 5x - 3 < 0
Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 9x + 4 ≥ 0
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Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)
Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x2+x−6)
Textbook Question
Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.
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