Domanda del libro di testo
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
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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 21
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.
Guida verificata passo dopo passo1
Start by rewriting the inequality: \(2x^{2} + 3x > 0\).
Factor the left-hand side expression: \(x(2x + 3) > 0\).
Identify the critical points by setting each factor equal to zero: \(x = 0\) and \(2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\).
Use the critical points to divide the real number line into intervals: \((-\infty, -\frac{3}{2})\), \((-\frac{3}{2}, 0)\), and \((0, \infty)\).
Test a value from each interval in the factored inequality \(x(2x + 3) > 0\) to determine where the product is positive, then express the solution set in interval notation.
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Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like >, <, ≥, or ≤. 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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06:07
Linear Inequalities
Factoring Quadratic Polynomials
Factoring quadratic polynomials means rewriting the expression as a product of two binomials or simpler polynomials. This step is crucial for solving inequalities because it helps identify the roots or zeros, which divide the number line into intervals to test for the inequality.
Video consigliato:
07:30Introduction to Factoring Polynomials
Interval Notation and Graphing on the Number Line
Interval notation is a concise way to represent sets of real numbers that satisfy inequalities, using parentheses and brackets to indicate open or closed intervals. Graphing the solution on a number line visually shows where the polynomial inequality holds true, aiding in understanding and verification.
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05:18Interval Notation
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