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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 89
Chapter 4, Problem 89

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=2x5−7x3+6x+8ƒ(x)=2x^5-7x^3+6x+8

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Identify the degree of the polynomial function \(f(x) = 2x^5 - 7x^3 + 6x + 8\). 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 \(f(x) = 2x^5 - 7x^3 + 6x + 8\). Each sign change corresponds to a possible positive zero, and the actual number of positive zeros is either equal to the number of sign changes or less than it by an even number.
Apply Descartes' Rule of Signs to \(f(-x)\) to find the possible number of negative real zeros. Substitute \(-x\) into the function to get \(f(-x) = 2(-x)^5 - 7(-x)^3 + 6(-x) + 8\), simplify it, and then count the sign changes. The number of negative zeros is either equal to the number of sign changes or less than it by an even number.
Determine the possible number of nonreal complex zeros by subtracting the possible numbers of positive and negative real zeros from the total degree (5). Remember that complex zeros come in conjugate pairs, so the number of nonreal complex zeros must be even.
Summarize the possible combinations of positive, negative, and nonreal complex zeros based on the results from the previous steps, ensuring the total number of zeros adds up to 5.

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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 roots total, 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 f(x) and f(-x). It provides an upper bound and possible variations in the count of positive and negative roots.
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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 roots is always even, which restricts the possible combinations of positive, negative, and nonreal zeros.
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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.

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

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]

Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=2x5−x4+x3−x2+x+5ƒ(x)=2x^5-x^4+x^3-x^2+x+5

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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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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)=x3+4x2-8x-8; [-3.8, -3]