Which product of prime polynomials is equivalent to 3x⁴ - 81x?
3x(x - 3)(x² - 3x - 9)
3x(x - 3)(x² + 3x + 9)
3x(x - 3)(x - 3)(x + 3)
3x(x - 3)(x + 3)(x + 3)

Solution:

Given polynimial is 3x⁴ - 81x

3x⁴ - 81x can be written as:

3x⁴ - 81x = 3x(x³ - 27)

From algebraic identities, we have 

a³ - b³ = (a - b)(a² + ab + b²)

Therefore

x³ - 27 = x³ - (3)³ = (x - 3)(x² + 3x + (3)² )

=(x - 3)(x² + 3x + 9)

3x⁴ - 81x = 3x(x³ - 27) = 3x(x - 3)(x² + 3x + 9)

Which product of prime polynomials is equivalent to 3x⁴ - 81x?

Summary:

3x(x - 3)(x² + 3x + 9) is the product of prime polynomials that is equivalent to 3x⁴ - 81x.