Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = 5 x^{6} - 6 x^{5} + 7 x^{3} - 4 x^{2} + x + 2$

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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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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.

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.

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.

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