Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
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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 20
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 writing the inequality: \(x^2 + 2x < 0\).
Factor the left-hand side expression: \(x^2 + 2x = x(x + 2)\).
Set the product less than zero: \(x(x + 2) < 0\). This means the product of two factors is negative.
Determine the critical points by setting each factor equal to zero: \(x = 0\) and \(x + 2 = 0 \Rightarrow x = -2\). These points divide the number line into intervals.
Test values from each interval \((-\infty, -2)\), \((-2, 0)\), and \((0, \infty)\) in the inequality \(x(x + 2) < 0\) to find where the product is negative, then express the solution set in interval notation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Polynomial Inequalities
Solving polynomial inequalities involves finding the values of the variable that make the inequality true. This typically requires factoring the polynomial, determining critical points where the expression equals zero, and testing intervals between these points to see where the inequality holds.
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Linear Inequalities
Factoring Quadratic Expressions
Factoring quadratic expressions means rewriting the quadratic as a product of two binomials. For example, x² + 2x can be factored as x(x + 2). Factoring helps identify the roots or zeros of the polynomial, which are essential for analyzing the inequality.
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Interval Notation and Number Line Graphing
Interval notation is a way to represent solution sets using intervals, such as (a, b) or [a, b), indicating which numbers satisfy the inequality. Graphing on a number line visually shows these intervals, helping to understand where the polynomial is positive or negative.
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Related Practice
Textbook Question
In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. 6x3+25x2−24x+5=0
Textbook Question
In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function.
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Textbook Question
In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x3−10x−12=0
Textbook Question
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
3
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the sum of y and w.