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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 41
Capitolo 4, Problema 41

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≥9x2x^3≥9x^2

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Start by rewriting the inequality \(x^3 \geq 9x^2\) so that one side is zero: subtract \$9x^2$ from both sides to get \(x^3 - 9x^2 \geq 0\).
Factor the left-hand side expression: factor out the greatest common factor \(x^2\), giving \(x^2(x - 9) \geq 0\).
Identify the critical points by setting each factor equal to zero: \(x^2 = 0\) gives \(x = 0\), and \(x - 9 = 0\) gives \(x = 9\). These points divide the number line into intervals to test.
Test the sign of the expression \(x^2(x - 9)\) in each interval determined by the critical points: \((-\infty, 0)\), \((0, 9)\), and \((9, \infty)\). Remember that \(x^2\) is always nonnegative, so the sign depends mainly on \((x - 9)\).
Based on the sign analysis, determine where the inequality \(x^2(x - 9) \geq 0\) holds true, include points where the expression equals zero, and express the solution set in interval notation. Finally, graph this solution set on the real number line.

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Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to 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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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 inequality problems.
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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 solution lies, using open or closed dots to indicate whether endpoints are included or excluded.
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