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.
