Final answer:
The query pertains to complex, real, and possible rational roots of a polynomial equation. However, the given equation seems to be flawed. The Fundamental Theorem of Algebra provides that a polynomial of degree n has exactly n complex roots, while the quadratic formula applies to quadratic equations.
Step-by-step explanation:
The equation presented is ¹⁷-2¹⁵-4¹⁵-2x-1=0, and we are asked to state the number of complex roots, the possible number of real roots, and the possible rational roots. Regrettably, there seems to be a misunderstanding as the original equation appears to be incorrect or incomplete. Typically, for a polynomial equation of degree n, the Fundamental Theorem of Algebra states that there will be exactly n complex roots (counting multiplicity). The number of real roots can vary but will also be at most n and will depend on the specific coefficients and terms of the equation. Rational roots, if any, can be found using the Rational Root Theorem, which states that any possible rational root is a divisor of the constant term divided by a divisor of the leading coefficient.
When dealing with a quadratic equation of the form ax²+bx+c=0, one can find the roots using the quadratic formula: x = (-b ± √(b²-4ac))/(2a). This will provide two solutions which could be real and/or complex depending on the discriminant (b²-4ac). If the discriminant is positive, there will be two distinct real roots, if it is zero, there will be exactly one real root (a repeated root), and if it is negative, there will be two complex roots.