Answer:
- potential: {±1/2, ±1, ±2, ±4}
- actual: [-2, -1, -1}
Step-by-step explanation:
The Rational Root Theorem tells you that potential rational roots are ...
±(divisor of constant term)/(divisor of leading coefficient)
Ignoring the fact that all of the coefficients can be reduced by a factor of 2, straightforward application of this theorem suggests possible rational roots might be ...
±{1, 2, 4}/{1, 2} = {±1/2, ±1, ±2, ±4} . . . potential roots
Examination of the graph of the function shows the actual roots are ...
x = -2, -1 (multiplicity 2)
_____
Additional comment
Descartes' rule of signs eliminates positive real roots from the list. (There are no coefficient sign changes, hence no positive real roots.)
Division of the polynomial by 2 gives x³ +4x² +5x +2 = 0, which only suggests rational roots of -2 or -1. The answer to the question of potential roots depends on the extent to which you refine the list above.
