Answer :

To simplify the expression \(\frac{8 x^3 - x^2 + 2 x - 1}{2 x}\), we can follow these steps:

1. Rewriting the expression:

We start by splitting the numerator into individual terms and then dividing each term by the denominator \(2x\):

[tex]\[ \frac{8 x^3}{2 x} - \frac{x^2}{2 x} + \frac{2 x}{2 x} - \frac{1}{2 x} \][/tex]

2. Simplifying each term individually:

- For the first term:
[tex]\[ \frac{8 x^3}{2 x} = \frac{8}{2} \cdot \frac{x^3}{x} = 4 x^2 \][/tex]

- For the second term:
[tex]\[ \frac{x^2}{2 x} = \frac{1}{2} \cdot \frac{x^2}{x} = \frac{x}{2} \][/tex]

- For the third term:
[tex]\[ \frac{2 x}{2 x} = 1 \][/tex]

- For the fourth term:
[tex]\[ \frac{1}{2 x} \][/tex]

3. Combining the simplified terms:

Putting all the simplified terms together, we have:

[tex]\[ 4 x^2 - \frac{x}{2} + 1 - \frac{1}{2 x} \][/tex]

Thus, the simplified form of the expression \(\frac{8 x^3 - x^2 + 2 x - 1}{2 x}\) is:

[tex]\[ 4 x^2 - \frac{x}{2} + 1 - \frac{1}{2 x} \][/tex]

This is the final simplified expression.