What is the product of prime polynomials that equals 3x^4 + 81x?

To determine the product of prime polynomials equivalent to the expression 3x⁴ + 81x, we start by factoring the expression.

First, notice that both terms share a common factor. The expression can be factored out as follows:

3x⁴ + 81x = 3x(x³ + 27)

Next, we can further factor the polynomial x³ + 27. This is a sum of cubes, which we can factor using the formula:

• a³ + b³ = (a + b)(a² – ab + b²)

In our case, a = x and b = 3, since 27 = 3³. Using the formula:

• (x + 3)(x² – 3x + 9)

Substituting this back into our expression gives us:

3x(x + 3)(x² – 3x + 9)

Therefore, the product of prime polynomials that is equivalent to 3x⁴ + 81x is:

• 3
• x
• (x + 3)
• (x² – 3x + 9)

In conclusion, we can write:

3x⁴ + 81x = 3x(x + 3)(x² – 3x + 9)