To determine if the polynomial x4 + 3x2 + x2 + 3 is prime, we first simplify it:
The polynomial can be rewritten as:
x4 + 4x2 + 3
Next, we check for possible factorizations. A prime polynomial cannot be factored into polynomials of lower degree with coefficients in the same field (real numbers, in this case).
We can use the Rational Root Theorem to check for possible rational roots, which might help indicate whether the polynomial can be factored. Testing values such as ±1, ±3, does not yield any rational roots.
Next, let’s examine if it’s reducible by looking for factorizations in the form (x² + ax + b)(x² + cx + d). However, through testing, we find that no integer values for a, b, c, and d satisfy this equation and result in coefficients that match our polynomial.
Finally, given that we cannot factor this polynomial into polynomials of lower degree and have found no rational roots, we conclude that:
The polynomial x4 + 4x2 + 3 is prime.