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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 7
Chapter 4, Problem 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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1
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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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 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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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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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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Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

Textbook Question

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. x2−6x+9<0

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Textbook Question

Find the domain of each rational function. f(x)=(x+7)/(x2+49)

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x12−3x2+5f(x)=x^{\(\frac\)12}-3x^2+5

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7