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)=x5−x3−1; between 1 and 2
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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 39
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, rewrite the inequality clearly: \(x^3 + x^2 + 4x + 4 > 0\).
Next, factor the cubic polynomial if possible. Try to factor by grouping: group terms as \((x^3 + x^2) + (4x + 4)\).
Factor out common factors from each group: \(x^2(x + 1) + 4(x + 1)\).
Since both groups contain the factor \((x + 1)\), factor it out: \((x + 1)(x^2 + 4) > 0\).
Analyze the factors separately: \(x + 1\) and \(x^2 + 4\). Note that \(x^2 + 4\) is always positive for all real \(x\) because \(x^2 \geq 0\) and adding 4 keeps it positive. So the inequality depends on the sign of \(x + 1\).
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Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs (>, <, ≥, ≤). 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 Polynomials
Factoring is the process of breaking down a polynomial into simpler polynomials (factors) whose product equals the original polynomial. This helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign.
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07:30Introduction 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, helping to interpret and communicate the solution clearly.
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05:18Interval Notation
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