When determining if a polynomial is prime, we check whether it can be factored into the product of two or more non-constant polynomials.
The given polynomial is: x³ + 3x³ + 2x + 6. First, we can simplify it by combining like terms:
x³ + 3x³ = 4x³. So, the polynomial simplifies to 4x³ + 2x + 6.
Now we can examine whether this polynomial can be factored. We can start by factoring out the common factor of 2:
2(2x³ + x + 3).
The remaining polynomial, 2x³ + x + 3, needs to be tested for primeness. Checking for possible rational roots via the Rational Root Theorem or attempting to factor by grouping shows that it does not factor neatly and does not have rational roots. Consequently, it does not break down into simpler polynomial factors.
Thus, the original polynomial can be expressed as a product of 2 (a constant) and 2x³ + x + 3, which means it is not prime. A prime polynomial cannot be expressed this way.
Therefore, the answer is that the polynomial is not prime as it can be factored into 2 and another polynomial.