Factor the following high-degree polynomial completely. 48x6−3x2

To factor the polynomial \( 48x^6 - 3x^2 \), we can start by factoring out the greatest common factor (GCF).

1. Identify the GCF:
- The coefficients are 48 and -3. The GCF of 48 and 3 is 3.
- The variable part has \( x^6 \) and \( x^2 \). The GCF for the variables is \( x^2 \).

So, the GCF of the entire polynomial is \( 3x^2 \).

2. Factor out the GCF from the polynomial:
\[
48x^6 - 3x^2 = 3x^2(16x^4 - 1)
\]

3. Now, we need to factor \( 16x^4 - 1 \). This expression is a difference of squares:
\[
16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 - 1)(4x^2 + 1)
\]

4. \( 4x^2 - 1 \) can be further factored, as it is also a difference of squares:
\[
4x^2 - 1 = (2x - 1)(2x + 1)
\]

5. The term \( 4x^2 + 1 \) cannot be factored over the real numbers, as it does not have real roots.

Putting it all together, the completely factored form of the polynomial \( 48x^6 - 3x^2 \) is:
\[
3x^2(2x - 1)(2x + 1)(4x^2 + 1)
\]

Thus, the final answer is:
\[
\boxed{3x^2(2x - 1)(2x + 1)(4x^2 + 1)}
\]