To determine which of the given polynomials is prime, we need to check if they can be factored into polynomials of lower degree with rational coefficients.
- x² + 9: This polynomial does not factor into real number components since it can be rewritten as x² + (3i)², which indicates it’s not a prime polynomial in the realm of real numbers.
- x² + 25: Similar to the first, this can be rewritten as x² + (5i)². Again, it doesn’t factor over the reals, thus not prime in this context.
- 3x² + 27: This can be factored as 3(x² + 9), showing that it can be expressed as a product of lower degree polynomials, hence, it is not prime.
- 2x² + 8: This also factors as 2(x² + 4), which indicates that it too is not a prime polynomial.
In conclusion, none of the given polynomials are prime as they can all be factored into products of lower degree polynomials. A polynomial is considered prime if it cannot be factored into the product of polynomials of lower degree with rational coefficients.