Question: Divide the following expression: $\frac{8x^6 - 6x^2 - 2x}{2x}$
1. **State the problem:** We need to divide the polynomial $8x^6 - 6x^2 - 2x$ by the monomial $2x$.
2. **Formula and rule:** When dividing a polynomial by a monomial, divide each term of the polynomial separately by the monomial:
$$\frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d}$$
3. **Apply division to each term:**
$$\frac{8x^6}{2x} - \frac{6x^2}{2x} - \frac{2x}{2x}$$
4. **Simplify each term:**
- For the first term:
$$\frac{8x^6}{2x} = \frac{\cancel{8}^4 \cancel{x^6}^{x^5}}{\cancel{2}^1 \cancel{x}^1} = 4x^5$$
- For the second term:
$$\frac{6x^2}{2x} = \frac{\cancel{6}^3 \cancel{x^2}^{x^1}}{\cancel{2}^1 \cancel{x}^1} = 3x$$
- For the third term:
$$\frac{2x}{2x} = \frac{\cancel{2}^1 \cancel{x}^1}{\cancel{2}^1 \cancel{x}^1} = 1$$
5. **Combine the simplified terms:**
$$4x^5 - 3x - 1$$
6. **Final answer:** The quotient is $4x^5 - 3x - 1$.
This matches option a.)