Textbook Question
Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x2 - 9x + 20 < 0
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 91
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
Verified step by step guidance1
Identify the degree of the polynomial function. Here, the degree is 6 because the highest power of \(x\) is 6 in \$5x^6$.
Use the Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) has exactly \(n\) roots (zeros) in the complex number system, counting multiplicities. So, there are 6 zeros in total.
Apply Descartes' Rule of Signs to determine the possible number of positive real zeros. Count the number of sign changes in the coefficients of \(f(x) = 5x^6 - 6x^5 + 7x^3 - 4x^2 + x + 2\).
Apply Descartes' Rule of Signs to \(f(-x)\) to determine the possible number of negative real zeros. Substitute \(-x\) into the function and count the sign changes in the resulting polynomial.
Use the total number of zeros (6) and the possible numbers of positive and negative real zeros to find the possible number of nonreal complex zeros by subtracting the sum of positive and negative zeros from 6.
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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. It ensures that the polynomial ƒ(x) = 5x^6 - 6x^5 + 7x^3 - 4x^2 + x + 2 has six roots, 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 ƒ(x) and ƒ(-x). It provides the maximum number of positive and negative roots and narrows down the possibilities for the zeros.
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Complex Conjugate Root Theorem
This theorem states that nonreal complex roots of polynomials with real coefficients occur in conjugate pairs. Therefore, the number of nonreal complex zeros must be even, which helps in determining the possible distribution of zeros for the given polynomial.
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Related Practice
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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Textbook Question
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
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Textbook Question
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=9x6-7x4+8x2+x+6
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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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