교과서 질문
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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4장, 문제 93
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
검증된 단계별 안내1
Identify the degree of the polynomial function \(f(x) = 7x^5 + 6x^4 + 2x^3 + 9x^2 + x + 5\). The degree is the highest power of \(x\), which is 5 in this case.
Use Descartes' Rule of Signs to determine the possible number of positive real zeros. Count the number of sign changes in the coefficients of \(f(x)\): \(7, +6, +2, +9, +1, +5\). Since all coefficients are positive, there are 0 sign changes, so there are 0 positive real zeros.
Apply Descartes' Rule of Signs to \(f(-x)\) to find the possible number of negative real zeros. Substitute \(-x\) into \(f(x)\) and simplify the signs of each term: \(7(-x)^5 + 6(-x)^4 + 2(-x)^3 + 9(-x)^2 + (-x) + 5 = -7x^5 + 6x^4 - 2x^3 + 9x^2 - x + 5\). Count the sign changes in the sequence \(-7, +6, -2, +9, -1, +5\).
Determine the possible number of negative real zeros by counting the sign changes in \(f(-x)\). Each sign change corresponds to a possible negative zero, and the actual number of negative zeros is either equal to the number of sign changes or less than it by an even number.
Use the Fundamental Theorem of Algebra to find the total number of zeros (counting multiplicities), which is equal to the degree (5). Subtract the possible number of positive and negative real zeros from 5 to find the possible number of nonreal complex zeros.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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) = 7x^5 + 6x^4 + 2x^3 + 9x^2 + x + 5 has five roots, which can be real or nonreal complex numbers.
추천 영상:
05:09
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 polynomial's zeros.
추천 영상:
6:54Cramer's Rule - 2 Equations with 2 Unknowns
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 real and nonreal roots for the given polynomial.
추천 영상:
05:33Complex Conjugates
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