To determine if a polynomial is prime, or irreducible, you need to check if it can be factored into the product of two non-constant polynomials with coefficients in the same field or ring.
Here are some basic steps to follow:
- Degree Check: First, check the degree of the polynomial. If it is a linear polynomial (degree 1), it is prime by definition.
- Factorization: For higher degrees, try to factor the polynomial. If you can express it as a product of lower-degree polynomials with integer or rational coefficients, then it is not prime.
- Use of Theorems: You can also use specific theorems like Eisenstein’s Criterion, which states that if a polynomial has specific properties concerning its coefficients and a prime number, then it is irreducible.
- Substitution: Check values of the polynomial at different integers. A non-zero value at some integer input indicates that the polynomial does not have a linear factor corresponding to that integer.
- Computer Algebra Systems: For complicated polynomials, using tools like computer algebra systems (CAS) can help. They can efficiently check for factors or provide irreducibility tests.
Ultimately, if you cannot find any factorization into non-constant polynomials, it’s likely that your polynomial is prime.