A polynomial is considered prime (or irreducible) if it cannot be factored into the product of two non-constant polynomials over the given field. In simpler terms, a prime polynomial is one that cannot be broken down into simpler polynomial factors.
To determine if the polynomial x⁴ + 3x² + x² + 3 is prime, we first simplify it. Combining the like terms:
- 3x² + x² = 4x²
So, the polynomial can be rewritten as:
x⁴ + 4x² + 3
Next, we can treat this polynomial as a quadratic in terms of x². By substituting y = x², we can rewrite it as:
y² + 4y + 3
Now, we can factor this quadratic polynomial:
(y + 1)(y + 3)
Substituting back x² for y, we get:
(x² + 1)(x² + 3)
This shows that the original polynomial x⁴ + 4x² + 3 can indeed be factored into the product of two non-constant polynomials, x² + 1 and x² + 3.
Therefore, since the polynomial can be factored, we conclude that:
The polynomial x⁴ + 3x² + x² + 3 (or x⁴ + 4x² + 3) is not prime.