Final answer:
The possible rational roots of the polynomial f(x) = 2x^4 - 5x^3 - 2x^2 - 3x + 8 are ±1, ±2, ±4, and ±8. These are found by listing the factors of the constant term over the factors of the leading coefficient per the Rational Root Theorem. Further work such as synthetic division would be needed to find which, if any, of these are actual roots of the polynomial.
Step-by-step explanation:
Finding Rational Roots Using The Rational Root Theorem
Let's examine the polynomial f(x) = 2x^4 - 5x^3 - 2x^2 - 3x + 8 and identify its possible rational roots using the Rational Root Theorem. According to the theorem, the potential rational roots are the factors of the constant term over the factors of the leading coefficient. In our case, the constant term is 8 and the leading coefficient is 2.
The factors of 8 are ±1, ±2, ±4, and ±8, and the factors of 2 are ±1 and ±2. Combining these factors, we can list all possible rational roots as ±1, ±2, ±4, ±8, ±1/2, ±2/2 (which simplifies to 1), ±4/2 (which simplifies to 2), and ±8/2 (which simplifies to 4). Therefore, the possible rational roots, removing the duplicates, are ±1, ±2, ±4, and ±8.
To identify the actual roots, one would typically use synthetic division or polynomial division to test these possible roots, checking to see which, if any, yield a zero remainder. The number of positive and negative real zeros can be estimated using Descartes' Rule of Signs, and complex zeros can be found using the Fundamental Theorem of Algebra, which states that a polynomial of degree n will have n roots (including real, negative, and complex) when counted with multiplicity.