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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 31
Chapter 4, Problem 31

Solve each polynomial inequality. Give the solution set in interval notation. -(x - 3)(x - 4)2 (x - 5) > 0

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First, rewrite the inequality clearly: \(-(x - 3)(x - 4)^2 (x - 5) > 0\).
Identify the critical points by setting each factor equal to zero: \(x - 3 = 0\), \(x - 4 = 0\), and \(x - 5 = 0\). These give \(x = 3\), \(x = 4\), and \(x = 5\).
Determine the sign of each factor in the intervals defined by the critical points: \((-\infty, 3)\), \((3, 4)\), \((4, 5)\), and \((5, \infty)\).
Consider the multiplicity of each factor: \((x - 4)^2\) has even multiplicity, so its sign does not change across \(x=4\). Also, remember the leading negative sign in front of the product affects the overall sign.
Test a sample value from each interval in the inequality to determine where the product is greater than zero, then express the solution set in interval notation based on these results.

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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 using inequality signs 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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Critical Points and Sign Analysis

Critical points are the values of the variable where the polynomial equals zero, found by setting each factor equal to zero. These points divide the number line into intervals, and testing the sign of the polynomial in each interval helps determine where the inequality holds.
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Multiplicity of Roots

The multiplicity of a root refers to how many times a factor appears in the polynomial. Even multiplicities cause the graph to touch the x-axis and not change sign at that root, while odd multiplicities cause the graph to cross the x-axis, changing the sign of the polynomial.
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