Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=2x5−x4+x3−x2+x+5ƒ(x)=$$2x^5-x^4$+x^3$-x^2+x+5$$

Accepted Answer

Identify the degree of the polynomial function $$f(x) = 2x^5$ - x^4$ + x^3$ - x^2$ + x + 5. $$The degree is the highest power of $$x$$, which is 5 in this case. Use Descartes' Rule of Signs to determine the possible number of positive real zeros. Count the number of sign changes in $$f(x)$$: $$2x^5 ($$positive), $$-x^4 ($$negative), $$+x^3 ($$positive), $$-x^2 ($$negative), $$+x ($$positive), $$+5 ($$positive). Each sign change indicates a possible positive zero or fewer by an even number. Apply Descartes' Rule of Signs to $$f(-x) to $$find the possible number of negative real zeros. Substitute $$-x$$ into the function and simplify the signs of each term, then count the sign changes in $$f(-x). $$Determine the possible number of nonreal complex zeros by using the fact that the total number of zeros (counting multiplicities) equals the degree of the polynomial. Subtract the possible positive and negative real zeros from the degree to find the number of nonreal complex zeros. Summarize the possible combinations of positive, negative, and nonreal complex zeros based on the counts from the previous steps, remembering that the number of zeros must add up to 5.

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

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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−x4+x3−x2+x+5ƒ(x)=2x^5-x^4+x^3-x^2+x+5