Textbook Question
Solve each problem. A comprehensive graph of ƒ(x)=x4-7x3+18x2-22x+12 is shown in the two screens, along with displays of the two real zeros. Find the two remaining nonreal complex zeros.
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 90
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x5-x3+7x-5
Verified step by step guidance1
Identify the degree of the polynomial function \(f(x) = 11x^5 - x^3 + 7x - 5\). The degree is 5, which means there are 5 zeros in total (counting multiplicities and including complex zeros).
Use Descartes' Rule of Signs to determine the possible number of positive real zeros. Count the number of sign changes in \(f(x) = 11x^5 - x^3 + 7x - 5\) by looking at the coefficients: \(+11\), \(-1\), \(+7\), \(-5\).
Apply Descartes' Rule of Signs to \(f(-x)\) to find the possible number of negative real zeros. Substitute \(-x\) into the function and simplify the signs of the terms, then count the sign changes in \(f(-x)\).
List all possible numbers of positive and negative real zeros based on the counts from steps 2 and 3, remembering that the number of zeros decreases by even numbers (e.g., if there are 3 sign changes, possible zeros are 3 or 1).
Determine the number of nonreal complex zeros by subtracting the total number of positive and negative real zeros from the degree 5, since the total number of zeros (real and complex) must equal the degree.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Algebra
This theorem states that a polynomial of degree n has exactly n roots in the complex number system, counting multiplicities. For the given fifth-degree polynomial, there are five zeros total, which can be real or nonreal complex numbers.
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Introduction to Algebraic Expressions
Descartes' Rule of Signs
Descartes' Rule of Signs helps determine the possible number of positive and negative real zeros of a polynomial by counting sign changes in f(x) and f(-x). It provides an upper bound on the number of positive and negative roots, aiding in identifying possible zero distributions.
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Complex Conjugate Root Theorem
This theorem states that nonreal complex zeros of polynomials with real coefficients occur in conjugate pairs. Therefore, the number of nonreal zeros must be even, which restricts the possible combinations of positive, negative, and nonreal zeros for the polynomial.
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Related Practice
Textbook Question
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
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Textbook Question
Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=x4-7x3+13x2+6x-28; [-1, 0]
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Textbook Question
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
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Textbook Question
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
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Textbook Question
Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x2 - x - 6 < 0
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