Final answer:
Determining the irreducibility of polynomials over Q[x] involves applying the Rational Root Theorem, Eisenstein's criterion, or other advanced techniques. The three given polynomials don't have obvious rational roots or meet the Eisenstein criterion, making their irreducibility over Q[x] not immediately clear without further analysis.
Step-by-step explanation:
Irreducibility of polynomials over the rational numbers Q[x] can be tested using various methods, including the Rational Root Theorem, Eisenstein's criterion, and by checking for factors with degrees less than the polynomial itself. Let's look at each polynomial given:
- x⁴ + x + 1: This does not have any rational roots, which can be tested quickly using the Rational Root Theorem. Therefore, we need more sophisticated methods, like Eisenstein's criterion, which is not applicable here, or a numerical method to conclude its irreducibility in Q[x].
- x³ − 2/3x² − 1/3: Similar to the first, the Rational Root Theorem doesn't yield any obvious rational roots. Further inspection or techniques might be required for a definitive answer.
- x⁶ + 15x⁵ − 30x³ + 6x + 120: This polynomial is more complex, but again none of the coefficients satisfy Eisenstein's criterion, and there are no rational roots. Its irreducibility is not immediately clear without more advanced techniques or factorization attempts.
To determine irreducibility, one might have to resort to factorization methods such as synthetic division to test potential factors or apply numerical methods for determining the presence of irrational or complex roots.
For more straightforward cases involving quadratics like x² + 0.0211x - 0.0211 = 0, we can apply the quadratic formula directly to find the roots.