Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = - 2 x^{5} + 10 x^{4} - 6 x^{3} + 8 x^{2} - x + 1$

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Identify the degree of the polynomial function. Since the highest power of $x$ is 5, the polynomial is of degree 5, so there are 5 zeros in total (counting multiplicities and complex zeros).

Use Descartes' Rule of Signs to determine the possible number of positive real zeros. Count the number of sign changes in $f(x) = -2x^5 + 10x^4 - 6x^3 + 8x^2 - x + 1$ by examining the signs of the coefficients in order.

Use Descartes' Rule of Signs to determine the possible number of negative real zeros by evaluating $f(-x)$ and counting the sign changes in the resulting polynomial.

List the possible numbers of positive and negative real zeros based on the counts from steps 2 and 3, remembering that the number of positive or negative zeros can be the number of sign changes or less than that by an even number (e.g., if 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 (including complex) must be 5.

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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. It ensures that the given fifth-degree polynomial has five zeros, 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, guiding the analysis of zero distribution.

Complex Conjugate Root Theorem

This theorem states that nonreal complex zeros of polynomials with real coefficients occur in conjugate pairs. It implies that the number of nonreal zeros is even, which helps in determining the possible counts 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. ƒ(x)=−2x5+10x4−6x3+8x2−x+1ƒ(x)=-2x^5+10x^4-6x^3+8x^2-x+1