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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 42
Capitolo 4, Problema 42

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. x3≤4x2

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Rewrite the inequality \(x^{3} \leq 4x^{2}\) by bringing all terms to one side to set the inequality to zero: \(x^{3} - 4x^{2} \leq 0\).
Factor the left-hand side expression: first, factor out the greatest common factor \(x^{2}\) to get \(x^{2}(x - 4) \leq 0\).
Identify the critical points by setting each factor equal to zero: \(x^{2} = 0\) gives \(x = 0\), and \(x - 4 = 0\) gives \(x = 4\). These points divide the real number line into intervals to test.
Test the sign of the expression \(x^{2}(x - 4)\) in each interval determined by the critical points: \((-\infty, 0)\), \((0, 4)\), and \((4, \infty)\), to determine where the inequality holds true.
Based on the sign test, write the solution set where \(x^{2}(x - 4) \leq 0\) is true, including points where the expression equals zero, and express the solution in interval notation.

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Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another expression using inequality signs (>, <, ≥, ≤). 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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Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials or factors. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.
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Introduction 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 inequality holds true, using open or closed dots to indicate whether endpoints are included.
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Interval Notation
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