What is the coefficient of x in the expression resulting from dividing 18x^3 by 12x^2 + 3x + 6x^2?

To find the coefficient of x in the division of the polynomial expression, we first need to clarify the operation. It appears that you are interested in dividing the polynomial expression 18x³ by the polynomial 12x² + 3x + 6x².

1. **Combine like terms** in the denominator:

• 12x² and 6x² combine to give 18x².
• So the denominator becomes: 18x² + 3x.

2. **Rewrite the expression** you seek to evaluate:

We are looking at:


{18x³} {18x² + 3x}

3. **Dividing 18x³ by 18x²** gives:

• This simplifies to:
• 18x³ / 18x² = x.

4. **Now, we must break down the remaining** 3x in the denominator to see how it interacts with x:

Continuing the polynomial long division:

1. The result of dividing 18x³} by 18x²} gives us x, as above.
2. Now, you multiply x by the entire denominator:

So next, we take:

• x * (18x² + 3x) = 18x³ + 3x².

5. **Subtracting this from the original numerator gives:**

• Original numerator: 18x³
• Minus the product: (18x³ + 3x²)
• Result: -3x².

6. This process continues, however, since we are mainly focused on the coefficient of x from our division, we find:

Initially from the division we found:

Coefficient of x = 1

Thus, the coefficient of x in the entire operation is simply 1.