To determine whether a polynomial is prime (also known as irreducible), we need to check if it can be factored into non-trivial polynomials with rational coefficients. In simpler terms, a polynomial is prime if it cannot be factored into lower-degree polynomials with rational coefficients.
Let's analyze each polynomial given:
1. [tex]\( p_1(x) = 3x^3 + 3x^2 - 2x - 2 \)[/tex]
2. [tex]\( p_2(x) = 3x^3 - 2x^2 + 3x - 4 \)[/tex]
3. [tex]\( p_3(x) = 4x^3 + 2x^2 + 6x + 3 \)[/tex]
4. [tex]\( p_4(x) = 4x^3 + 4x^2 - 3x - 3 \)[/tex]
### Checking Each Polynomial:
1. For [tex]\( 3x^3 + 3x^2 - 2x - 2 \)[/tex]:
- To check the irreducibility of [tex]\( 3x^3 + 3x^2 - 2x - 2 \)[/tex], we attempt to factor the polynomial. If we find two non-trivial factors (i.e., not simply [tex]\( 1 \)[/tex] and the polynomial itself), then the polynomial is not irreducible. Upon inspection and testing, this polynomial can indeed be factored into lower-degree polynomials with rational coefficients.
2. For [tex]\( 3x^3 - 2x^2 + 3x - 4 \)[/tex]:
- Similarly, we attempt to factor [tex]\( 3x^3 - 2x^2 + 3x - 4 \)[/tex]. Attempting various factorization methods shows that this polynomial can be factored as well.
3. For [tex]\( 4x^3 + 2x^2 + 6x + 3 \)[/tex]:
- We apply the same factorization approach to [tex]\( 4x^3 + 2x^2 + 6x + 3 \)[/tex]. Upon further inspection, this polynomial can also be factored into lower-degree polynomials with rational coefficients.
4. For [tex]\( 4x^3 + 4x^2 - 3x - 3 \)[/tex]:
- Finally, we test [tex]\( 4x^3 + 4x^2 - 3x - 3 \)[/tex] for factorability. Testing various factorization methods confirms that this polynomial can be factored into polynomials with rational coefficients as well.
### Conclusion:
We find that all four polynomials [tex]\( 3x^3 + 3x^2 - 2x - 2 \)[/tex], [tex]\( 3x^3 - 2x^2 + 3x - 4 \)[/tex], [tex]\( 4x^3 + 2x^2 + 6x + 3 \)[/tex], and [tex]\( 4x^3 + 4x^2 - 3x - 3 \)[/tex] can be factored into lower-degree polynomials with rational coefficients.
Therefore, none of the given polynomials is a prime (irreducible) polynomial. The result is:
- [tex]\( 3x^3 + 3x^2 - 2x - 2 \)[/tex]: Not prime (False)
- [tex]\( 3x^3 - 2x^2 + 3x - 4 \)[/tex]: Not prime (False)
- [tex]\( 4x^3 + 2x^2 + 6x + 3 \)[/tex]: Not prime (False)
- [tex]\( 4x^3 + 4x^2 - 3x - 3 \)[/tex]: Not prime (False)
So, the answer is that no polynomial among the given ones is prime.
