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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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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주요 개념

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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.

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.

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.

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