Textbook Question
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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Table of contents
- 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
Chapter 4, Problem 37
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x4+5x3+5x2−5x−6;f(3)
Verified step by step guidance1
Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = x^{4} + 5x^{3} + 5x^{2} - 5x - 6\) and you want to find \(f(3)\).
Set up synthetic division by writing down the coefficients of the polynomial: \(1\) (for \(x^{4}\)), \(5\) (for \(x^{3}\)), \(5\) (for \(x^{2}\)), \(-5\) (for \(x\)), and \(-6\) (constant term).
Write the value \(3\) (the input for \(f(3)\)) to the left of the synthetic division setup.
Perform synthetic division by bringing down the first coefficient, then multiply it by \(3\), add this result to the next coefficient, and repeat this process across all coefficients.
The final number you obtain after completing synthetic division is the remainder, which equals \(f(3)\) by the Remainder Theorem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making it faster and less error-prone. This method helps find the quotient and remainder efficiently.
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Remainder Theorem
The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means evaluating the polynomial at x = c gives the remainder directly, which is useful for quickly finding function values or checking factors.
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Polynomial Evaluation
Polynomial evaluation involves calculating the value of a polynomial function at a specific input x = c. Using synthetic division or direct substitution, you can find f(c), which represents the output of the polynomial for that input. This is essential for understanding function behavior and solving related problems.
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Related Practice
Textbook Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x4−5x3−x2−6x+4
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Textbook Question
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x4−5x3−x2+3x+2; f(−1/2)
Textbook Question
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3+x2−2x+1; between -3 and -2
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.
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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)=3x3−10x+9; between -3 and -2
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