College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Simplify the following expression:

[tex]
\[
\frac{4+8x^6-12x^2}{4x^2}
\]
[/tex]

a) [tex]\( x^4 + 2x^3 - 3x^2 \)[/tex]

b) [tex]\( x^6 + 2x^4 - 8x^2 \)[/tex]

c) [tex]\( x^2 + 2x^3 - 3 \)[/tex]

d) [tex]\( x^6 + 2x^4 - 3 \)[/tex]

Answer :

Sure, let's simplify the given expression step-by-step:

The expression is:
[tex]\[
\frac{4 + 8x^6 - 12x^2}{4x^2}
\][/tex]

1. Distribute the denominator to each term in the numerator:
[tex]\[
\frac{4}{4x^2} + \frac{8x^6}{4x^2} - \frac{12x^2}{4x^2}
\][/tex]

2. Simplify each fraction:
- For the first term:
[tex]\[
\frac{4}{4x^2} = \frac{4}{4} \cdot \frac{1}{x^2} = 1 \cdot \frac{1}{x^2} = x^{-2}
\][/tex]
- For the second term:
[tex]\[
\frac{8x^6}{4x^2} = \frac{8}{4} \cdot \frac{x^6}{x^2} = 2 \cdot x^{6-2} = 2x^4
\][/tex]
- For the third term:
[tex]\[
\frac{12x^2}{4x^2} = \frac{12}{4} \cdot \frac{x^2}{x^2} = 3 \cdot 1 = 3
\][/tex]

3. Combine the simplified terms:
[tex]\[
x^{-2} + 2x^4 - 3
\][/tex]

So, the simplified form of the given expression is:
[tex]\[
2x^4 - 3 + x^{-2}
\][/tex]

Among the given options, this matches with the correctly formatted answer. Therefore, the correct choice is:
```
2x4 - 3 + x(-2)
```

Therefore, the correct answer is:
```
2
x^4 - 3 + x^(-2)
```