Which polynomial is prime?
x³ + 3x² - 2x - 6
x³ - 2x² + 3x - 6
4x⁴ + 4x³ - 2x - 2
2x⁴ + x3 - x + 2

Solution:

A prime polynomial has only two factors 1 and itself. It is a polynomial with integer coefficients that cannot be factored into polynomials of lower degrees.

To find the prime polynomial, we will factorize all the polynomials.

Equation 1: x³ + 3x² - 2x - 6 can be factored into (x + 3) (x² - 2). Therefore, it is not a prime polynomial.

Equation 2: x³ - 2x² + 3x - 6 can be factored into (x- 2) (x² + 3). Therefore, it is not a prime polynomial.

Equation 3: 4x⁴ + 4x³ - 2x - 2 can be factored into 2(x + 1) (x³ - 1). Therefore, it is not a prime polynomial.

Equation 4: 2x⁴ + x³ - x + 2 can not be factorized with rational numbers. Therefore, it is a prime polynomial.

Which polynomial is prime? x³ + 3x² - 2x - 6, x³ - 2x² + 3x - 6, 4x⁴ + 4x³ - 2x - 2, 2x⁴ + x3 - x + 2

Summary:

The prime polynomial that cannot be factored into a polynomial of lower degree is 2x⁴ + x³ - x + 2.