To determine which polynomial is prime, we first need to understand what it means for a polynomial to be prime. A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with coefficients in the same field (in this context, typically the field of real numbers or integers).
Let's examine each given polynomial to see if it meets this condition:
1. Polynomial: [tex]\(x^3 + 3x^2 + 2x + 6\)[/tex]
- To check if this polynomial is prime, we need to attempt to factor it.
- Upon attempting to factor [tex]\(x^3 + 3x^2 + 2x + 6\)[/tex], it turns out that it is not prime since it can be factored into the product of lower-degree polynomials over the real numbers.
2. Polynomial: [tex]\(x^3 + 3x^2 - 2x - 6\)[/tex]
- Similarly, we will attempt to factor this polynomial.
- Factoring [tex]\(x^3 + 3x^2 - 2x - 6\)[/tex], results in the realization that it can indeed be factored into lower-degree polynomials, indicating it is not prime.
3. Polynomial: [tex]\(10x^2 - 4x + 3x + 6\)[/tex]
- First, let's simplify the polynomial: [tex]\(10x^2 - 4x + 3x + 6\)[/tex] simplifies to [tex]\(10x^2 - x + 6\)[/tex].
- We then check if [tex]\(10x^2 - x + 6\)[/tex] can be factored further.
- Upon inspection and factoring attempts, [tex]\(10x^2 - x + 6\)[/tex] cannot be factored into the product of non-constant polynomials with integer coefficients or real coefficients. Hence, it remains irreducible and is considered a prime polynomial.
4. Polynomial: [tex]\(10x^2 - 10x + 6x - 6\)[/tex]
- Simplify this polynomial: [tex]\(10x^2 - 10x + 6x - 6\)[/tex] simplifies to [tex]\(10x^2 - 4x - 6\)[/tex].
- We now check for its factorability.
- After factoring attempts, [tex]\(10x^2 - 4x - 6\)[/tex] can be factored into non-constant polynomials, thus it is not prime.
After examining all four polynomials, we determine that:
- The polynomial [tex]\(10x^2 - 4x + 3x + 6\)[/tex] simplifies to [tex]\(10x^2 - x + 6\)[/tex] and is the only polynomial that is prime and cannot be factored further.
Hence, the polynomial [tex]\(10x^2 - 4x + 3x + 6\)[/tex] is the prime polynomial.
