What is the remainder when we multiply 3×4, 2×3, x2, 2x, and 9×2 together?

To find the remainder when multiplying the expression 3x^4, 2x^3, x^2, 2x, and 9x^2, we will first calculate the product of these terms and then determine the remainder of the total product.

Let’s break down the steps:

1. Start with the given expressions: 3x⁴, 2x³, x², 2x, and 9x².
2. The total product can be expressed as:
• 3x⁴ * 2x³ * x² * 2x * 9x²
3. Next, let’s simplify this product by grouping like terms:
• 3 * 2 * 1 * 2 * 9 gives us a coefficient of 108.
• For the variable parts, we will add the exponents:
• x⁴ * x³ * x² * x¹ * x² results in x^{(4 + 3 + 2 + 1 + 2) = x12}
4. Thus, the total product simplifies to 108x¹².
5. To find the remainder when dividing by a number, we typically want a divisor. However, since the problem does not specify one, we cannot calculate a numerical remainder with only the expression itself.
6. If we are to express these terms in modulo arithmetic without a specific modulus, we conclude that the total expression is dependent on any arbitrary chosen divisor for a specific calculation.

In conclusion, without additional context on a divisor, the result simplifies to 108x¹², and further computation of a remainder would require a defined modulus.