Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5
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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 11
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. 3x2+10x−8≤0
Verified step by step guidance1
Start by writing down the inequality: \(3x^{2} + 10x - 8 \leq 0\).
To solve the inequality, first find the roots of the corresponding quadratic equation \(3x^{2} + 10x - 8 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=3\), \(b=10\), and \(c=-8\).
Calculate the discriminant \(\Delta = b^{2} - 4ac = 10^{2} - 4 \times 3 \times (-8)\) to determine the nature of the roots.
Find the two roots by substituting the values into the quadratic formula. These roots will divide the real number line into intervals.
Test a value from each interval in the original inequality \(3x^{2} + 10x - 8 \leq 0\) to determine which intervals satisfy the inequality, 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 Quadratic Polynomials
Factoring is the process of rewriting a quadratic polynomial as a product of two binomials. This helps identify the roots (zeros) of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.
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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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Related Practice
Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (x4−81)/(x−3)
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Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As ______
Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z.
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. 9x2+3x−2≥0
Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (4x4−4x2+6x)/(x−4)