Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=-2x^5+10x^4-6x^3+8x^2-x+1

Verified step by step guidance

1. Identify the degree of the polynomial function, which is the highest power of x. In this case, the degree is 5.
2. Apply the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots, counting multiplicities and including complex roots.
3. 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).
4. To find the possible number of negative real zeros, apply Descartes' Rule of Signs to f(-x) by substituting -x for x in the polynomial and counting the sign changes.
5. Consider the possibility of nonreal complex zeros. Remember that complex zeros occur in conjugate pairs, so the number of nonreal complex zeros must be even.

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n has exactly n roots in the complex number system, counting multiplicities. This means that for a polynomial like ƒ(x) = -2x^5 + 10x^4 - 6x^3 + 8x^2 - x + 1, there will be five roots, which can be real or complex.

Descarte's Rule of Signs

Descarte's Rule of Signs provides a method to determine the number of positive and negative real roots of a polynomial by analyzing the sign changes in the coefficients. For positive roots, count the sign changes in ƒ(x), and for negative roots, evaluate ƒ(-x) and count the sign changes. This helps in predicting the nature of the roots.

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients, any nonreal complex roots must occur in conjugate pairs. This means if a + bi is a root, then a - bi is also a root. This theorem is essential for understanding the distribution of roots when analyzing polynomials with real coefficients.