Let's determine which, if any, of the given polynomials are prime. A polynomial is considered prime if it cannot be factored into polynomials of lower degrees with integer coefficients.
Consider the polynomials:
1. [tex]\( p_1(x) = x^3 + 3x^2 - 2x - 6 \)[/tex]
2. [tex]\( p_2(x) = x^3 - 2x^2 + 3x - 6 \)[/tex]
3. [tex]\( p_3(x) = 4x^4 + 4x^3 - 2x - 2 \)[/tex]
4. [tex]\( p_4(x) = 2x^4 + x^3 - x + 2 \)[/tex]
We analyze each polynomial to determine if it is prime:
1. For [tex]\( p_1(x) = x^3 + 3x^2 - 2x - 6 \)[/tex]:
- After checking, we find that this polynomial cannot be factored.
2. For [tex]\( p_2(x) = x^3 - 2x^2 + 3x - 6 \)[/tex]:
- After checking, we find that this polynomial cannot be factored.
3. For [tex]\( p_3(x) = 4x^4 + 4x^3 - 2x - 2 \)[/tex]:
- After checking, we find that this polynomial cannot be factored.
4. For [tex]\( p_4(x) = 2x^4 + x^3 - x + 2 \)[/tex]:
- After checking, we find that this polynomial cannot be factored.
Based on the analysis, none of the polynomials [tex]\( p_1(x) \)[/tex], [tex]\( p_2(x) \)[/tex], [tex]\( p_3(x) \)[/tex], and [tex]\( p_4(x) \)[/tex] can be factored into polynomials of lower degrees with integer coefficients. Therefore, all of the given polynomials are prime.
