Final answer:
The possible rational roots of the polynomial f(p)=2p³+4p²+p-4 are determined by the Rational Root Theorem, which are ±1, ±2, ±4, -1, -2, -4, ±1/2, and -1/2.
Step-by-step explanation:
The question asks us to find all the possible rational roots of the given polynomial f(p) = 2p³ + 4p² + p - 4. To do this, we can use the Rational Root Theorem, which states that any rational root of the polynomial when written in lowest terms p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. For our polynomial, the constant term is -4, and the leading coefficient is 2.
First, let's list the factors of the constant term -4, which are ±1, ±2, and ±4. Next, we'll list the factors of the leading coefficient 2, which are ±1 and ±2. Considering all combinations of these factors, and taking both positive and negative versions, gives us possible values for p and q. Our possible rational roots (p/q) are thus ±1/1, ±2/1, ±4/1, -1/1, -2/1, and -4/1, as well as ±1/2, ±2/2, ±4/2, -1/2, -2/2, and -4/2.
After simplifying, we have a set of potential rational roots: ±1, ±2, ±4, -1, -2, -4, ±1/2, and -1/2 (note that ±2/2 and ±4/2 simplify to 1 and 2, respectively, which are already listed, and -2/2 and -4/2 simplify to -1 and -2, which are also already listed). These are all the possible rational roots of the polynomial f(p). To determine whether any of these are actual roots, we would need to substitute them into the polynomial and see if the result is zero. Therefore, the complete set of possible rational roots for f(p) = 2p³ + 4p² + p - 4 is 1, 2, 4, -1, -2, -4, 1/2, and -1/2.