- #1
Poirot1
- 245
- 0
Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.
Polynomial theorems provide a set of rules and criteria for determining whether a polynomial is irreducible. By applying these theorems to the polynomial x^4 −7, we can determine if it is irreducible or not.
The first step is to factor the polynomial into its irreducible components. For x^4 −7, this would be (x^2+√7)(x^2-√7).
Yes, polynomial theorems can be used for polynomials of any degree. However, the complexity of the theorems may increase with higher degrees.
Some commonly used polynomial theorems include the rational root theorem, the Eisenstein criterion, and the irreducibility criteria for quadratic and cubic polynomials.
Yes, it is possible for a polynomial to satisfy the criteria of one theorem but not another. This is why it is important to use multiple theorems to determine the irreducibility of a polynomial.