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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 7
Capitolo 4, Problema 7

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. x2+5x+4>0

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Start by rewriting the inequality: \(x^2 + 5x + 4 > 0\).
Factor the quadratic expression on the left side: \(x^2 + 5x + 4 = (x + 1)(x + 4)\).
Identify the critical points by setting each factor equal to zero: \(x + 1 = 0\) gives \(x = -1\), and \(x + 4 = 0\) gives \(x = -4\).
Use the critical points to divide the real number line into intervals: \((-\infty, -4)\), \((-4, -1)\), and \((-1, \infty)\).
Test a value from each interval in the inequality \((x + 1)(x + 4) > 0\) 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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Linear Inequalities

Factoring Quadratic Polynomials

Factoring is the process of expressing a quadratic polynomial as a product of two binomials. For example, x² + 5x + 4 factors to (x + 1)(x + 4). Factoring helps identify the roots of the polynomial, which are critical points for determining where the polynomial changes sign.
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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 is positive or negative. Using critical points from factoring, intervals are tested to determine where the inequality holds true.
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Interval Notation
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