Domanda del libro di testo
In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x4+6x3−18x2; between 2 and 3
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 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.
Guida verificata passo dopo passo1
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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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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06:07
Linear Inequalities
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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07:30Introduction to Factoring Polynomials
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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05:18Interval Notation
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