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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 35
Chapter 4, Problem 35

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+2x2−x−2≥0x^3+2x^2−x−2≥0

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First, write down the inequality clearly: \(x^3 + 2x^2 - x - 2 \geq 0\).
Next, factor the cubic polynomial on the left side. Start by grouping terms: \((x^3 + 2x^2) - (x + 2)\).
Factor each group: \(x^2(x + 2) - 1(x + 2)\), then factor out the common binomial: \((x + 2)(x^2 - 1)\).
Recognize that \(x^2 - 1\) is a difference of squares and factor it further: \((x + 2)(x - 1)(x + 1)\).
Set each factor equal to zero to find critical points: \(x + 2 = 0\), \(x - 1 = 0\), and \(x + 1 = 0\), which gives \(x = -2\), \(x = 1\), and \(x = -1\). Use these points to divide the number line into intervals and test the sign of the polynomial in each interval to determine where the inequality holds.

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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 (>, <, ≥, ≤). 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 breaking down a polynomial into simpler polynomials (factors) whose product equals the original polynomial. This step is crucial for solving polynomial inequalities because it helps identify the roots, which divide the number line into intervals to test for the inequality.
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Interval Notation and Sign Analysis

Interval notation is a way to express solution sets using intervals on the real number line. After finding roots, sign analysis involves testing values in each interval to determine where the polynomial satisfies the inequality. The solution is then expressed using interval notation to clearly show all valid values.
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Related Practice
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. (1−x)2(x−5/2)<0

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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)=(x2+4x−21)/(x+7)

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Textbook Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x2+4x−3

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Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3−4x2+2; between 0 and 1

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Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x3−7x2−2x+5;f(−3)

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