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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 26
Chapter 4, Problem 26

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≤2x+2

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1
Rewrite the inequality so that all terms are on one side, resulting in a standard polynomial inequality form: \(x^2 - 2x - 2 \leq 0\).
Find the roots of the quadratic equation \(x^2 - 2x - 2 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=-2\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots and then find the exact roots.
Use the roots to divide the real number line into intervals. Test a point from each interval in the inequality \(x^2 - 2x - 2 \leq 0\) to determine where the inequality holds true.
Express the solution set as an interval or union of intervals based on the test results, and then graph this solution set on the real number line.

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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 another value using inequality symbols (e.g., ≤, <, >, ≥). Solving them requires finding all 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 and Solving Quadratic Equations

To solve polynomial inequalities, especially quadratics, it is essential to rewrite the inequality in standard form and factor the polynomial if possible. Factoring helps identify critical points (roots) where the expression equals zero, which divide the number line into intervals for testing.
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Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality holds, solutions are expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing these intervals on a real number line visually shows the solution set, indicating included or excluded endpoints.
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Related Practice
Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x2+4)

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Textbook Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91

Textbook Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=3(x+5)(x+2)2

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Textbook Question

Divide using synthetic division. (x2−5x−5x3+x4)÷(5+x)

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In Exercises 25–26, graph each polynomial function. f(x)=−x3(x+4)2(x−1)f(x) = -x^3(x + 4)^2(x-1)

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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. 9x2−6x+1<0

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