Certainly! Let's solve the expression [tex]\(\frac{x^4 - 2x^2 - 3}{x^2}\)[/tex] step-by-step.
1. Write the expression:
[tex]\[
\frac{x^4 - 2x^2 - 3}{x^2}
\][/tex]
2. Separate the terms in the numerator:
[tex]\[
\frac{x^4}{x^2} - \frac{2x^2}{x^2} - \frac{3}{x^2}
\][/tex]
3. Simplify each term individually:
- The first term is [tex]\(\frac{x^4}{x^2}\)[/tex]:
[tex]\[
x^4 \div x^2 = x^{4-2} = x^2
\][/tex]
- The second term is [tex]\(\frac{2x^2}{x^2}\)[/tex]:
[tex]\[
2x^2 \div x^2 = 2
\][/tex]
- The third term is [tex]\(\frac{3}{x^2}\)[/tex]:
This term remains [tex]\(\frac{3}{x^2}\)[/tex] as it cannot be simplified further.
4. Combine the simplified terms:
[tex]\[
x^2 - 2 - \frac{3}{x^2}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{x^4 - 2x^2 - 3}{x^2} = x^2 - 2 - \frac{3}{x^2}
\][/tex]
