Domanda del libro di testo
Identify which graphs are not those of polynomial functions.
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 4, Problema 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
Guida verificata passo dopo passo1
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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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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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
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