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A. What are the possible rational roots of the polynomial \(f(x) = 2x^4 - 5x^3 - 2x^2 - 3x + 8\)?

a) ±1, ±2, ±4, ±8
b) ±1, ±2, ±3, ±4
c) ±1, ±4, ±8, ±16
d) ±1, ±3, ±5, ±7

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Final answer:

The rational roots of the polynomial 2x^4 - 5x^3 - 2x^2 - 3x + 8 can be found using the Rational Root Theorem, leading to possible rational roots of ±(1, 1/2, 2, 4, 8), corresponding to option a) ±(1, ±2, ±4, ±8).

Step-by-step explanation:

The rational roots of a polynomial can be predicted using the Rational Root Theorem, which states the possible rational roots are of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In the polynomial f(x) = 2x^4 - 5x^3 - 2x^2 - 3x + 8, the constant term is 8 and the leading coefficient is 2. Therefore, the possible values for p are ±(1, 2, 4, 8) and the possible values for q are ±(1, 2).

When we list out the possible p/q combinations, we eliminate duplicates and get the set of possible rational roots: ±(1, 1/2, 2, 4, 4/2, 8, 8/2). Simplifying the fractions, we end up with ±(1, 1/2, 2, 4, 8), which corresponds to option a) ±(1, ±2, ±4, ±8).

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