In mathematics, a prime polynomial is one that cannot be factored into the product of two non-constant polynomials. To determine which of the given polynomials are prime, we need to examine each one individually:
- x³: This polynomial is a cubic polynomial and cannot be factored into polynomials of lower degree that are non-constant (e.g., it cannot be expressed as a product of two polynomials like (x + a)(x² + b), where a and b are constants). Therefore, x³ is a prime polynomial.
- 3x³: This polynomial can be factored as 3 * x³. Although 3 is a constant, it can still be expressed as the product of two factors (3 and x³), which means it is not prime. Therefore, 3x³ is not a prime polynomial.
- 2x: Similar to 3x³, the polynomial 2x can be factored as 2 * x, meaning it consists of constant and non-constant factors. Thus, 2x is not a prime polynomial.
- 6: The number 6 is a constant polynomial. It can be factored into 2 * 3, hence it can be expressed as the product of two factors. Therefore, 6 is not a prime polynomial.
In conclusion, among the polynomials provided, only x³ qualifies as a prime polynomial.