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

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. According to the Fundamental Theorem of Algebra, 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. Apply Descartes' Rule of Signs to f(-x) to determine the possible number of negative real zeros. Substitute -x for x in the function and count the sign changes.
5. Calculate the possible number of nonreal complex zeros by subtracting the total number of real zeros (positive and negative) from the degree of the polynomial.

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^5-7x^3+6x+8, 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 zeros of a polynomial by analyzing the sign changes in the function's coefficients. For positive zeros, count the sign changes in ƒ(x), and for negative zeros, evaluate ƒ(-x) and count the sign changes there.

Complex Zeros

Complex zeros occur in conjugate pairs for polynomials with real coefficients. If a polynomial has nonreal complex zeros, they will appear as pairs of the form a + bi and a - bi. Understanding this concept is crucial for determining the total number of zeros, as any remaining zeros after accounting for real zeros must be complex.