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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 91
Chapter 4, Problem 91

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=5x6−6x5+7x3−4x2+x+2ƒ(x)=5x^6-6x^5+7x^3-4x^2+x+2

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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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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

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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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.

Textbook Question

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 2x2 - 9x ≥ 18

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

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

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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