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 12
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
Verified step by step guidance1
Start by writing down the inequality: \(9x^2 + 3x - 2 \geq 0\).
Find the roots of the quadratic equation \(9x^2 + 3x - 2 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=9\), \(b=3\), and \(c=-2\).
Calculate the discriminant \(\Delta = b^2 - 4ac = 3^2 - 4 \times 9 \times (-2)\) to determine the nature of the roots.
Use the roots found to divide the real number line into intervals. Test a value from each interval in the original inequality \(9x^2 + 3x - 2 \geq 0\) to determine where the inequality holds true.
Express the solution set as the union of intervals where the inequality is satisfied, and write the solution in interval notation. Then, graph these intervals on a 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 expressing a quadratic polynomial as a product of two binomials. This helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequality problems.
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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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Related Practice
Textbook Question
Identify which graphs are not those of polynomial functions.
Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x2−8x+3
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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
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