Problems 90-93 are based on material learned earlier in the...

Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Use Descartes' Rule of Signs to determine the possible number of positive or negative real zeros for the function $f(x)=-2 x^3+6 x^2-7 x-8$.

Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Use Descartes' Rule of Signs to determine the possible number of positive or negative real zeros for the function $f(x)=-2 x^3+6 x^2-7 x-8$.

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Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a method used in algebra to estimate the number of positive and negative real zeros of a polynomial. It works by counting the number of sign changes in the sequence of nonzero coefficients of the polynomial. For positive zeros, the rule directly applies to the given polynomial; for negative zeros, the rule is applied to the polynomial with x replaced by -x. The actual number of positive or negative real zeros is either equal to the number of sign changes or less than that number by an even integer.

Polynomial Functions

Polynomial functions are algebraic expressions that involve variables raised to nonnegative integer powers and are combined using addition, subtraction, and multiplication. The zeros of a polynomial, which are the solutions to the equation f(x)=0, are critical for understanding the function's behavior. Determining the number and nature of these zeros, including their real and complex components, is an essential part of algebra and precalculus, often facilitated by tools like Descartes' Rule of Signs.

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