Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=2x3−4x2+2x+7ƒ(x)=$$2x^3$-4x^2+2x+7$$

Accepted Answer

Identify the degree of the polynomial function $$f(x) = 2x^3$ - 4x^2$ + 2x + 7. $$Since the highest power of $$x is 3$$, the degree is 3, so there are 3 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 $$f(x) = 2x^3$ - 4x^2$ + 2x + 7. $$Each sign change corresponds to 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 to get $$f(-x) = 2(-x)^3$ - 4(-x)^2$ + 2(-x) + 7$$, simplify it, and count the sign changes in this new polynomial. Determine the possible number of nonreal complex zeros by subtracting the possible number of positive and negative real zeros from the total degree. Remember that complex zeros come in conjugate pairs, so the number of nonreal zeros must be even. Summarize the possible combinations of positive, negative, and nonreal zeros based on the results from the previous steps, ensuring the total number of zeros adds up to 3.

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

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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)=2x3−4x2+2x+7ƒ(x)=2x^3-4x^2+2x+7