To simplify the expression [tex]\( y = \frac{-4 - x^3}{2x^2} \)[/tex], let's proceed through a detailed, step-by-step solution.
1. Identify the expression to simplify:
[tex]\[
y = \frac{-4 - x^3}{2x^2}
\][/tex]
2. Separate the numerator:
We can rewrite the fraction by breaking up the numerator over the common denominator:
[tex]\[
y = \frac{-4}{2x^2} + \frac{-x^3}{2x^2}
\][/tex]
3. Simplify each term individually:
- For the first term, [tex]\(\frac{-4}{2x^2}\)[/tex]:
[tex]\[
\frac{-4}{2x^2} = \frac{-4}{2} \cdot \frac{1}{x^2} = -2 \cdot \frac{1}{x^2} = \frac{-2}{x^2}
\][/tex]
- For the second term, [tex]\(\frac{-x^3}{2x^2}\)[/tex]:
[tex]\[
\frac{-x^3}{2x^2} = -\frac{x^3}{2x^2}
\][/tex]
We can simplify the power of [tex]\( x \)[/tex]:
[tex]\[
\frac{x^3}{x^2} = x^{3-2} = x^1 = x
\][/tex]
So, we have:
[tex]\[
-\frac{x^3}{2x^2} = -\frac{x}{2}
\][/tex]
4. Combine the simplified terms:
Putting both simplified terms together, we get:
[tex]\[
y = \frac{-2}{x^2} - \frac{x}{2}
\][/tex]
Therefore, the simplified form of the expression [tex]\( y = \frac{-4 - x^3}{2x^2} \)[/tex] is:
[tex]\[
y = \frac{-2}{x^2} - \frac{x}{2}
\][/tex]
