Final answer:
The equation x⁵ − x⁴ − x³ − 5x² − 12x − 6 = 0 has no rational roots and must have either two or four irrational roots.
Step-by-step explanation:
To show that the equation x⁵ − x⁴ − x³ − 5x² − 12x − 6 = 0 has exactly one rational root, we can use the Rational Root Theorem. According to the theorem, any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-6) and q is a factor of the leading coefficient (1). In this case, the possible rational roots are ±1, ±2, ±3, and ±6. By substituting each of these values into the equation, we find that none of them satisfy the equation. Therefore, there is no rational root.
To prove that the equation must have either two or four irrational roots, we can use the Fundamental Theorem of Algebra. The theorem states that a polynomial equation of degree n has exactly n roots, including both real and complex roots. Since the equation x⁵ − x⁴ − x³ − 5x² − 12x − 6 = 0 is a polynomial equation of degree 5, it must have exactly 5 roots. We already know that there is one rational root, so the remaining roots must be irrational. Therefore, the equation must have either two or four irrational roots.