Question

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)=$$-2x^5+10x^4$-6x^3$+8x^2-x+1$$

Accepted Answer

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.

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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Key Concepts

Fundamental Theorem of Algebra

Descartes' Rule of Signs

Complex Conjugate Root Theorem

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